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## G = Dic10⋊3D6order 480 = 25·3·5

### 3rd semidirect product of Dic10 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic10⋊3D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×D12 — Dic10⋊3D6
 Lower central C15 — C30 — C60 — Dic10⋊3D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic103D6
G = < a,b,c,d | a20=c6=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, cbc-1=a10b, dbd=a5b, dcd=c-1 >

Subgroups: 892 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], C12, C12 [×2], D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], C3⋊C8, C3⋊C8, D12, D12 [×2], C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C4.Dic3, D4⋊S3, D4⋊S3, Q82S3 [×2], C2×D12, C3×C4○D4, C3×Dic5, C3×Dic5, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D4⋊D6, C5×C3⋊C8, C153C8, C5⋊D12, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×D12, D60, D4×C15, C2×S3×D5, D8⋊D5, C20.32D6, C20.D6, C15⋊SD16, C5×D4⋊S3, D4⋊D15, D5×D12, C3×D42D5, Dic103D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D4⋊D6, C2×S3×D5, D8⋊D5, D5×C3⋊D4, Dic103D6

Smallest permutation representation of Dic103D6
On 120 points
Generators in S120
(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)
(1 106 11 116)(2 105 12 115)(3 104 13 114)(4 103 14 113)(5 102 15 112)(6 101 16 111)(7 120 17 110)(8 119 18 109)(9 118 19 108)(10 117 20 107)(21 81 31 91)(22 100 32 90)(23 99 33 89)(24 98 34 88)(25 97 35 87)(26 96 36 86)(27 95 37 85)(28 94 38 84)(29 93 39 83)(30 92 40 82)(41 75 51 65)(42 74 52 64)(43 73 53 63)(44 72 54 62)(45 71 55 61)(46 70 56 80)(47 69 57 79)(48 68 58 78)(49 67 59 77)(50 66 60 76)
(1 95 51)(2 84 52 10 96 60)(3 93 53 19 97 49)(4 82 54 8 98 58)(5 91 55 17 99 47)(6 100 56)(7 89 57 15 81 45)(9 87 59 13 83 43)(11 85 41)(12 94 42 20 86 50)(14 92 44 18 88 48)(16 90 46)(21 71 110 23 69 112)(22 80 111 32 70 101)(24 78 113 30 72 119)(25 67 114 39 73 108)(26 76 115 28 74 117)(27 65 116 37 75 106)(29 63 118 35 77 104)(31 61 120 33 79 102)(34 68 103 40 62 109)(36 66 105 38 64 107)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)

G:=sub<Sym(120)| (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), (1,106,11,116)(2,105,12,115)(3,104,13,114)(4,103,14,113)(5,102,15,112)(6,101,16,111)(7,120,17,110)(8,119,18,109)(9,118,19,108)(10,117,20,107)(21,81,31,91)(22,100,32,90)(23,99,33,89)(24,98,34,88)(25,97,35,87)(26,96,36,86)(27,95,37,85)(28,94,38,84)(29,93,39,83)(30,92,40,82)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,70,56,80)(47,69,57,79)(48,68,58,78)(49,67,59,77)(50,66,60,76), (1,95,51)(2,84,52,10,96,60)(3,93,53,19,97,49)(4,82,54,8,98,58)(5,91,55,17,99,47)(6,100,56)(7,89,57,15,81,45)(9,87,59,13,83,43)(11,85,41)(12,94,42,20,86,50)(14,92,44,18,88,48)(16,90,46)(21,71,110,23,69,112)(22,80,111,32,70,101)(24,78,113,30,72,119)(25,67,114,39,73,108)(26,76,115,28,74,117)(27,65,116,37,75,106)(29,63,118,35,77,104)(31,61,120,33,79,102)(34,68,103,40,62,109)(36,66,105,38,64,107), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)>;

G:=Group( (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), (1,106,11,116)(2,105,12,115)(3,104,13,114)(4,103,14,113)(5,102,15,112)(6,101,16,111)(7,120,17,110)(8,119,18,109)(9,118,19,108)(10,117,20,107)(21,81,31,91)(22,100,32,90)(23,99,33,89)(24,98,34,88)(25,97,35,87)(26,96,36,86)(27,95,37,85)(28,94,38,84)(29,93,39,83)(30,92,40,82)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,70,56,80)(47,69,57,79)(48,68,58,78)(49,67,59,77)(50,66,60,76), (1,95,51)(2,84,52,10,96,60)(3,93,53,19,97,49)(4,82,54,8,98,58)(5,91,55,17,99,47)(6,100,56)(7,89,57,15,81,45)(9,87,59,13,83,43)(11,85,41)(12,94,42,20,86,50)(14,92,44,18,88,48)(16,90,46)(21,71,110,23,69,112)(22,80,111,32,70,101)(24,78,113,30,72,119)(25,67,114,39,73,108)(26,76,115,28,74,117)(27,65,116,37,75,106)(29,63,118,35,77,104)(31,61,120,33,79,102)(34,68,103,40,62,109)(36,66,105,38,64,107), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93) );

G=PermutationGroup([(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)], [(1,106,11,116),(2,105,12,115),(3,104,13,114),(4,103,14,113),(5,102,15,112),(6,101,16,111),(7,120,17,110),(8,119,18,109),(9,118,19,108),(10,117,20,107),(21,81,31,91),(22,100,32,90),(23,99,33,89),(24,98,34,88),(25,97,35,87),(26,96,36,86),(27,95,37,85),(28,94,38,84),(29,93,39,83),(30,92,40,82),(41,75,51,65),(42,74,52,64),(43,73,53,63),(44,72,54,62),(45,71,55,61),(46,70,56,80),(47,69,57,79),(48,68,58,78),(49,67,59,77),(50,66,60,76)], [(1,95,51),(2,84,52,10,96,60),(3,93,53,19,97,49),(4,82,54,8,98,58),(5,91,55,17,99,47),(6,100,56),(7,89,57,15,81,45),(9,87,59,13,83,43),(11,85,41),(12,94,42,20,86,50),(14,92,44,18,88,48),(16,90,46),(21,71,110,23,69,112),(22,80,111,32,70,101),(24,78,113,30,72,119),(25,67,114,39,73,108),(26,76,115,28,74,117),(27,65,116,37,75,106),(29,63,118,35,77,104),(31,61,120,33,79,102),(34,68,103,40,62,109),(36,66,105,38,64,107)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,60),(8,59),(9,58),(10,57),(11,56),(12,55),(13,54),(14,53),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93)])

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 12E 15A 15B 20A 20B 30A 30B 30C 30D 30E 30F 40A 40B 40C 40D 60A 60B order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 6 6 8 8 10 10 10 10 10 10 12 12 12 12 12 15 15 20 20 30 30 30 30 30 30 40 40 40 40 60 60 size 1 1 4 10 12 60 2 2 10 20 2 2 2 4 4 20 12 60 2 2 8 8 24 24 4 10 10 20 20 4 4 4 4 4 4 8 8 8 8 12 12 12 12 8 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 C8⋊C22 S3×D5 D4×D5 D4⋊D6 C2×S3×D5 D8⋊D5 D5×C3⋊D4 Dic10⋊3D6 kernel Dic10⋊3D6 C20.32D6 C20.D6 C15⋊SD16 C5×D4⋊S3 D4⋊D15 D5×D12 C3×D4⋊2D5 D4⋊2D5 C3×Dic5 C6×D5 D4⋊S3 Dic10 C4×D5 C5×D4 C3⋊C8 D12 C3×D4 Dic5 D10 C15 D4 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 1 2 2 2 2 4 4 2

Matrix representation of Dic103D6 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 52 1 240 188 0 0 240 190 52 51 0 0 233 125 0 240 0 0 234 124 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 16 0 148 225 0 0 164 0 148 77 0 0 15 169 0 0 0 0 200 57 148 225
,
 0 240 0 0 0 0 1 240 0 0 0 0 0 0 51 1 0 0 0 0 51 190 0 0 0 0 0 0 1 0 0 0 52 1 189 240
,
 1 240 0 0 0 0 0 240 0 0 0 0 0 0 52 1 188 240 0 0 240 190 51 52 0 0 0 0 240 0 0 0 51 1 240 0

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,240,233,234,0,0,1,190,125,124,0,0,240,52,0,0,0,0,188,51,240,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,16,164,15,200,0,0,0,0,169,57,0,0,148,148,0,148,0,0,225,77,0,225],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,51,51,0,52,0,0,1,190,0,1,0,0,0,0,1,189,0,0,0,0,0,240],[1,0,0,0,0,0,240,240,0,0,0,0,0,0,52,240,0,51,0,0,1,190,0,1,0,0,188,51,240,240,0,0,240,52,0,0] >;

Dic103D6 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_3D_6
% in TeX

G:=Group("Dic10:3D6");
// GroupNames label

G:=SmallGroup(480,554);
// by ID

G=gap.SmallGroup(480,554);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^6=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^10*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

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