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## G = C2×Q8.10D14order 448 = 26·7

### Direct product of C2 and Q8.10D14

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×Q8.10D14
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7 — C2×Q8×D7 — C2×Q8.10D14
 Lower central C7 — C14 — C2×Q8.10D14
 Upper central C1 — C22 — C22×Q8

Generators and relations for C2×Q8.10D14
G = < a,b,c,d,e | a2=b4=1, c2=d14=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d13 >

Subgroups: 2868 in 794 conjugacy classes, 447 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22×Q8, C22×Q8, C2×C4○D4, 2- 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×Q8, C22×D7, C22×C14, C2×2- 1+4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, Q8×D7, Q82D7, C2×C7⋊D4, C22×C28, Q8×C14, C2×C4○D28, C2×Q8×D7, C2×Q82D7, Q8.10D14, Q8×C2×C14, C2×Q8.10D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2- 1+4, C25, C22×D7, C2×2- 1+4, C23×D7, Q8.10D14, D7×C24, C2×Q8.10D14

Smallest permutation representation of C2×Q8.10D14
On 224 points
Generators in S224
(1 169)(2 170)(3 171)(4 172)(5 173)(6 174)(7 175)(8 176)(9 177)(10 178)(11 179)(12 180)(13 181)(14 182)(15 183)(16 184)(17 185)(18 186)(19 187)(20 188)(21 189)(22 190)(23 191)(24 192)(25 193)(26 194)(27 195)(28 196)(29 126)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 137)(41 138)(42 139)(43 140)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 121)(53 122)(54 123)(55 124)(56 125)(57 156)(58 157)(59 158)(60 159)(61 160)(62 161)(63 162)(64 163)(65 164)(66 165)(67 166)(68 167)(69 168)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 155)(85 218)(86 219)(87 220)(88 221)(89 222)(90 223)(91 224)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 204)(100 205)(101 206)(102 207)(103 208)(104 209)(105 210)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)
(1 209 15 223)(2 210 16 224)(3 211 17 197)(4 212 18 198)(5 213 19 199)(6 214 20 200)(7 215 21 201)(8 216 22 202)(9 217 23 203)(10 218 24 204)(11 219 25 205)(12 220 26 206)(13 221 27 207)(14 222 28 208)(29 77 43 63)(30 78 44 64)(31 79 45 65)(32 80 46 66)(33 81 47 67)(34 82 48 68)(35 83 49 69)(36 84 50 70)(37 57 51 71)(38 58 52 72)(39 59 53 73)(40 60 54 74)(41 61 55 75)(42 62 56 76)(85 192 99 178)(86 193 100 179)(87 194 101 180)(88 195 102 181)(89 196 103 182)(90 169 104 183)(91 170 105 184)(92 171 106 185)(93 172 107 186)(94 173 108 187)(95 174 109 188)(96 175 110 189)(97 176 111 190)(98 177 112 191)(113 163 127 149)(114 164 128 150)(115 165 129 151)(116 166 130 152)(117 167 131 153)(118 168 132 154)(119 141 133 155)(120 142 134 156)(121 143 135 157)(122 144 136 158)(123 145 137 159)(124 146 138 160)(125 147 139 161)(126 148 140 162)
(1 139 15 125)(2 126 16 140)(3 113 17 127)(4 128 18 114)(5 115 19 129)(6 130 20 116)(7 117 21 131)(8 132 22 118)(9 119 23 133)(10 134 24 120)(11 121 25 135)(12 136 26 122)(13 123 27 137)(14 138 28 124)(29 184 43 170)(30 171 44 185)(31 186 45 172)(32 173 46 187)(33 188 47 174)(34 175 48 189)(35 190 49 176)(36 177 50 191)(37 192 51 178)(38 179 52 193)(39 194 53 180)(40 181 54 195)(41 196 55 182)(42 183 56 169)(57 85 71 99)(58 100 72 86)(59 87 73 101)(60 102 74 88)(61 89 75 103)(62 104 76 90)(63 91 77 105)(64 106 78 92)(65 93 79 107)(66 108 80 94)(67 95 81 109)(68 110 82 96)(69 97 83 111)(70 112 84 98)(141 217 155 203)(142 204 156 218)(143 219 157 205)(144 206 158 220)(145 221 159 207)(146 208 160 222)(147 223 161 209)(148 210 162 224)(149 197 163 211)(150 212 164 198)(151 199 165 213)(152 214 166 200)(153 201 167 215)(154 216 168 202)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 146 15 160)(2 159 16 145)(3 144 17 158)(4 157 18 143)(5 142 19 156)(6 155 20 141)(7 168 21 154)(8 153 22 167)(9 166 23 152)(10 151 24 165)(11 164 25 150)(12 149 26 163)(13 162 27 148)(14 147 28 161)(29 88 43 102)(30 101 44 87)(31 86 45 100)(32 99 46 85)(33 112 47 98)(34 97 48 111)(35 110 49 96)(36 95 50 109)(37 108 51 94)(38 93 52 107)(39 106 53 92)(40 91 54 105)(41 104 55 90)(42 89 56 103)(57 173 71 187)(58 186 72 172)(59 171 73 185)(60 184 74 170)(61 169 75 183)(62 182 76 196)(63 195 77 181)(64 180 78 194)(65 193 79 179)(66 178 80 192)(67 191 81 177)(68 176 82 190)(69 189 83 175)(70 174 84 188)(113 220 127 206)(114 205 128 219)(115 218 129 204)(116 203 130 217)(117 216 131 202)(118 201 132 215)(119 214 133 200)(120 199 134 213)(121 212 135 198)(122 197 136 211)(123 210 137 224)(124 223 138 209)(125 208 139 222)(126 221 140 207)

G:=sub<Sym(224)| (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,193)(26,194)(27,195)(28,196)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,140)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,124)(56,125)(57,156)(58,157)(59,158)(60,159)(61,160)(62,161)(63,162)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(91,224)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217), (1,209,15,223)(2,210,16,224)(3,211,17,197)(4,212,18,198)(5,213,19,199)(6,214,20,200)(7,215,21,201)(8,216,22,202)(9,217,23,203)(10,218,24,204)(11,219,25,205)(12,220,26,206)(13,221,27,207)(14,222,28,208)(29,77,43,63)(30,78,44,64)(31,79,45,65)(32,80,46,66)(33,81,47,67)(34,82,48,68)(35,83,49,69)(36,84,50,70)(37,57,51,71)(38,58,52,72)(39,59,53,73)(40,60,54,74)(41,61,55,75)(42,62,56,76)(85,192,99,178)(86,193,100,179)(87,194,101,180)(88,195,102,181)(89,196,103,182)(90,169,104,183)(91,170,105,184)(92,171,106,185)(93,172,107,186)(94,173,108,187)(95,174,109,188)(96,175,110,189)(97,176,111,190)(98,177,112,191)(113,163,127,149)(114,164,128,150)(115,165,129,151)(116,166,130,152)(117,167,131,153)(118,168,132,154)(119,141,133,155)(120,142,134,156)(121,143,135,157)(122,144,136,158)(123,145,137,159)(124,146,138,160)(125,147,139,161)(126,148,140,162), (1,139,15,125)(2,126,16,140)(3,113,17,127)(4,128,18,114)(5,115,19,129)(6,130,20,116)(7,117,21,131)(8,132,22,118)(9,119,23,133)(10,134,24,120)(11,121,25,135)(12,136,26,122)(13,123,27,137)(14,138,28,124)(29,184,43,170)(30,171,44,185)(31,186,45,172)(32,173,46,187)(33,188,47,174)(34,175,48,189)(35,190,49,176)(36,177,50,191)(37,192,51,178)(38,179,52,193)(39,194,53,180)(40,181,54,195)(41,196,55,182)(42,183,56,169)(57,85,71,99)(58,100,72,86)(59,87,73,101)(60,102,74,88)(61,89,75,103)(62,104,76,90)(63,91,77,105)(64,106,78,92)(65,93,79,107)(66,108,80,94)(67,95,81,109)(68,110,82,96)(69,97,83,111)(70,112,84,98)(141,217,155,203)(142,204,156,218)(143,219,157,205)(144,206,158,220)(145,221,159,207)(146,208,160,222)(147,223,161,209)(148,210,162,224)(149,197,163,211)(150,212,164,198)(151,199,165,213)(152,214,166,200)(153,201,167,215)(154,216,168,202), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,146,15,160)(2,159,16,145)(3,144,17,158)(4,157,18,143)(5,142,19,156)(6,155,20,141)(7,168,21,154)(8,153,22,167)(9,166,23,152)(10,151,24,165)(11,164,25,150)(12,149,26,163)(13,162,27,148)(14,147,28,161)(29,88,43,102)(30,101,44,87)(31,86,45,100)(32,99,46,85)(33,112,47,98)(34,97,48,111)(35,110,49,96)(36,95,50,109)(37,108,51,94)(38,93,52,107)(39,106,53,92)(40,91,54,105)(41,104,55,90)(42,89,56,103)(57,173,71,187)(58,186,72,172)(59,171,73,185)(60,184,74,170)(61,169,75,183)(62,182,76,196)(63,195,77,181)(64,180,78,194)(65,193,79,179)(66,178,80,192)(67,191,81,177)(68,176,82,190)(69,189,83,175)(70,174,84,188)(113,220,127,206)(114,205,128,219)(115,218,129,204)(116,203,130,217)(117,216,131,202)(118,201,132,215)(119,214,133,200)(120,199,134,213)(121,212,135,198)(122,197,136,211)(123,210,137,224)(124,223,138,209)(125,208,139,222)(126,221,140,207)>;

G:=Group( (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,193)(26,194)(27,195)(28,196)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,140)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,124)(56,125)(57,156)(58,157)(59,158)(60,159)(61,160)(62,161)(63,162)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(91,224)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217), (1,209,15,223)(2,210,16,224)(3,211,17,197)(4,212,18,198)(5,213,19,199)(6,214,20,200)(7,215,21,201)(8,216,22,202)(9,217,23,203)(10,218,24,204)(11,219,25,205)(12,220,26,206)(13,221,27,207)(14,222,28,208)(29,77,43,63)(30,78,44,64)(31,79,45,65)(32,80,46,66)(33,81,47,67)(34,82,48,68)(35,83,49,69)(36,84,50,70)(37,57,51,71)(38,58,52,72)(39,59,53,73)(40,60,54,74)(41,61,55,75)(42,62,56,76)(85,192,99,178)(86,193,100,179)(87,194,101,180)(88,195,102,181)(89,196,103,182)(90,169,104,183)(91,170,105,184)(92,171,106,185)(93,172,107,186)(94,173,108,187)(95,174,109,188)(96,175,110,189)(97,176,111,190)(98,177,112,191)(113,163,127,149)(114,164,128,150)(115,165,129,151)(116,166,130,152)(117,167,131,153)(118,168,132,154)(119,141,133,155)(120,142,134,156)(121,143,135,157)(122,144,136,158)(123,145,137,159)(124,146,138,160)(125,147,139,161)(126,148,140,162), (1,139,15,125)(2,126,16,140)(3,113,17,127)(4,128,18,114)(5,115,19,129)(6,130,20,116)(7,117,21,131)(8,132,22,118)(9,119,23,133)(10,134,24,120)(11,121,25,135)(12,136,26,122)(13,123,27,137)(14,138,28,124)(29,184,43,170)(30,171,44,185)(31,186,45,172)(32,173,46,187)(33,188,47,174)(34,175,48,189)(35,190,49,176)(36,177,50,191)(37,192,51,178)(38,179,52,193)(39,194,53,180)(40,181,54,195)(41,196,55,182)(42,183,56,169)(57,85,71,99)(58,100,72,86)(59,87,73,101)(60,102,74,88)(61,89,75,103)(62,104,76,90)(63,91,77,105)(64,106,78,92)(65,93,79,107)(66,108,80,94)(67,95,81,109)(68,110,82,96)(69,97,83,111)(70,112,84,98)(141,217,155,203)(142,204,156,218)(143,219,157,205)(144,206,158,220)(145,221,159,207)(146,208,160,222)(147,223,161,209)(148,210,162,224)(149,197,163,211)(150,212,164,198)(151,199,165,213)(152,214,166,200)(153,201,167,215)(154,216,168,202), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,146,15,160)(2,159,16,145)(3,144,17,158)(4,157,18,143)(5,142,19,156)(6,155,20,141)(7,168,21,154)(8,153,22,167)(9,166,23,152)(10,151,24,165)(11,164,25,150)(12,149,26,163)(13,162,27,148)(14,147,28,161)(29,88,43,102)(30,101,44,87)(31,86,45,100)(32,99,46,85)(33,112,47,98)(34,97,48,111)(35,110,49,96)(36,95,50,109)(37,108,51,94)(38,93,52,107)(39,106,53,92)(40,91,54,105)(41,104,55,90)(42,89,56,103)(57,173,71,187)(58,186,72,172)(59,171,73,185)(60,184,74,170)(61,169,75,183)(62,182,76,196)(63,195,77,181)(64,180,78,194)(65,193,79,179)(66,178,80,192)(67,191,81,177)(68,176,82,190)(69,189,83,175)(70,174,84,188)(113,220,127,206)(114,205,128,219)(115,218,129,204)(116,203,130,217)(117,216,131,202)(118,201,132,215)(119,214,133,200)(120,199,134,213)(121,212,135,198)(122,197,136,211)(123,210,137,224)(124,223,138,209)(125,208,139,222)(126,221,140,207) );

G=PermutationGroup([[(1,169),(2,170),(3,171),(4,172),(5,173),(6,174),(7,175),(8,176),(9,177),(10,178),(11,179),(12,180),(13,181),(14,182),(15,183),(16,184),(17,185),(18,186),(19,187),(20,188),(21,189),(22,190),(23,191),(24,192),(25,193),(26,194),(27,195),(28,196),(29,126),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,137),(41,138),(42,139),(43,140),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,121),(53,122),(54,123),(55,124),(56,125),(57,156),(58,157),(59,158),(60,159),(61,160),(62,161),(63,162),(64,163),(65,164),(66,165),(67,166),(68,167),(69,168),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151),(81,152),(82,153),(83,154),(84,155),(85,218),(86,219),(87,220),(88,221),(89,222),(90,223),(91,224),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(99,204),(100,205),(101,206),(102,207),(103,208),(104,209),(105,210),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217)], [(1,209,15,223),(2,210,16,224),(3,211,17,197),(4,212,18,198),(5,213,19,199),(6,214,20,200),(7,215,21,201),(8,216,22,202),(9,217,23,203),(10,218,24,204),(11,219,25,205),(12,220,26,206),(13,221,27,207),(14,222,28,208),(29,77,43,63),(30,78,44,64),(31,79,45,65),(32,80,46,66),(33,81,47,67),(34,82,48,68),(35,83,49,69),(36,84,50,70),(37,57,51,71),(38,58,52,72),(39,59,53,73),(40,60,54,74),(41,61,55,75),(42,62,56,76),(85,192,99,178),(86,193,100,179),(87,194,101,180),(88,195,102,181),(89,196,103,182),(90,169,104,183),(91,170,105,184),(92,171,106,185),(93,172,107,186),(94,173,108,187),(95,174,109,188),(96,175,110,189),(97,176,111,190),(98,177,112,191),(113,163,127,149),(114,164,128,150),(115,165,129,151),(116,166,130,152),(117,167,131,153),(118,168,132,154),(119,141,133,155),(120,142,134,156),(121,143,135,157),(122,144,136,158),(123,145,137,159),(124,146,138,160),(125,147,139,161),(126,148,140,162)], [(1,139,15,125),(2,126,16,140),(3,113,17,127),(4,128,18,114),(5,115,19,129),(6,130,20,116),(7,117,21,131),(8,132,22,118),(9,119,23,133),(10,134,24,120),(11,121,25,135),(12,136,26,122),(13,123,27,137),(14,138,28,124),(29,184,43,170),(30,171,44,185),(31,186,45,172),(32,173,46,187),(33,188,47,174),(34,175,48,189),(35,190,49,176),(36,177,50,191),(37,192,51,178),(38,179,52,193),(39,194,53,180),(40,181,54,195),(41,196,55,182),(42,183,56,169),(57,85,71,99),(58,100,72,86),(59,87,73,101),(60,102,74,88),(61,89,75,103),(62,104,76,90),(63,91,77,105),(64,106,78,92),(65,93,79,107),(66,108,80,94),(67,95,81,109),(68,110,82,96),(69,97,83,111),(70,112,84,98),(141,217,155,203),(142,204,156,218),(143,219,157,205),(144,206,158,220),(145,221,159,207),(146,208,160,222),(147,223,161,209),(148,210,162,224),(149,197,163,211),(150,212,164,198),(151,199,165,213),(152,214,166,200),(153,201,167,215),(154,216,168,202)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,146,15,160),(2,159,16,145),(3,144,17,158),(4,157,18,143),(5,142,19,156),(6,155,20,141),(7,168,21,154),(8,153,22,167),(9,166,23,152),(10,151,24,165),(11,164,25,150),(12,149,26,163),(13,162,27,148),(14,147,28,161),(29,88,43,102),(30,101,44,87),(31,86,45,100),(32,99,46,85),(33,112,47,98),(34,97,48,111),(35,110,49,96),(36,95,50,109),(37,108,51,94),(38,93,52,107),(39,106,53,92),(40,91,54,105),(41,104,55,90),(42,89,56,103),(57,173,71,187),(58,186,72,172),(59,171,73,185),(60,184,74,170),(61,169,75,183),(62,182,76,196),(63,195,77,181),(64,180,78,194),(65,193,79,179),(66,178,80,192),(67,191,81,177),(68,176,82,190),(69,189,83,175),(70,174,84,188),(113,220,127,206),(114,205,128,219),(115,218,129,204),(116,203,130,217),(117,216,131,202),(118,201,132,215),(119,214,133,200),(120,199,134,213),(121,212,135,198),(122,197,136,211),(123,210,137,224),(124,223,138,209),(125,208,139,222),(126,221,140,207)]])

94 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2M 4A ··· 4L 4M ··· 4T 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 14 ··· 14 2 ··· 2 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4

94 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D7 D14 D14 2- 1+4 Q8.10D14 kernel C2×Q8.10D14 C2×C4○D28 C2×Q8×D7 C2×Q8⋊2D7 Q8.10D14 Q8×C2×C14 C22×Q8 C22×C4 C2×Q8 C14 C2 # reps 1 6 4 4 16 1 3 9 36 2 12

Matrix representation of C2×Q8.10D14 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 24 13 0 0 0 0 16 5 0 0 5 16 0 0 0 0 13 24 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 27 11 0 0 0 0 18 2 0 0 0 0 0 0 2 18 0 0 0 0 11 27
,
 19 10 0 0 0 0 19 7 0 0 0 0 0 0 0 0 26 8 0 0 0 0 21 21 0 0 3 21 0 0 0 0 8 8 0 0
,
 28 22 0 0 0 0 0 1 0 0 0 0 0 0 7 20 0 0 0 0 12 22 0 0 0 0 0 0 7 20 0 0 0 0 12 22

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,13,0,0,0,0,16,24,0,0,24,16,0,0,0,0,13,5,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,2,11,0,0,0,0,18,27],[19,19,0,0,0,0,10,7,0,0,0,0,0,0,0,0,3,8,0,0,0,0,21,8,0,0,26,21,0,0,0,0,8,21,0,0],[28,0,0,0,0,0,22,1,0,0,0,0,0,0,7,12,0,0,0,0,20,22,0,0,0,0,0,0,7,12,0,0,0,0,20,22] >;

C2×Q8.10D14 in GAP, Magma, Sage, TeX

C_2\times Q_8._{10}D_{14}
% in TeX

G:=Group("C2xQ8.10D14");
// GroupNames label

G:=SmallGroup(448,1374);
// by ID

G=gap.SmallGroup(448,1374);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^14=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^13>;
// generators/relations

׿
×
𝔽