Copied to
clipboard

## G = C2×D10.D6order 480 = 25·3·5

### Direct product of C2 and D10.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D10.D6
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C2×C3⋊F5 — C22×C3⋊F5 — C2×D10.D6
 Lower central C15 — C30 — C2×D10.D6
 Upper central C1 — C22 — C23

Generators and relations for C2×D10.D6
G = < a,b,c,d,e | a2=b10=c2=d6=1, e2=b4c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b7, cd=dc, ece-1=b6c, ede-1=b5d-1 >

Subgroups: 1324 in 264 conjugacy classes, 81 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C23, D5, D5, C10, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, F5, D10, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C22×C6, C22×C6, C3×D5, C3×D5, C30, C30, C30, C2×C22⋊C4, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C6.D4, C22×Dic3, C23×C6, C3⋊F5, C6×D5, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C22⋊F5, C22×F5, C23×D5, C2×C6.D4, C2×C3⋊F5, C2×C3⋊F5, D5×C2×C6, D5×C2×C6, D5×C2×C6, C22×C30, C2×C22⋊F5, D10.D6, C22×C3⋊F5, D5×C22×C6, C2×D10.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, F5, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C2×F5, C6.D4, C22×Dic3, C2×C3⋊D4, C3⋊F5, C22⋊F5, C22×F5, C2×C6.D4, C2×C3⋊F5, C2×C22⋊F5, D10.D6, C22×C3⋊F5, C2×D10.D6

Smallest permutation representation of C2×D10.D6
On 120 points
Generators in S120
(1 87)(2 88)(3 89)(4 90)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 77)(22 78)(23 79)(24 80)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 26)(22 25)(23 24)(27 30)(28 29)(31 38)(32 37)(33 36)(34 35)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 52)(53 60)(54 59)(55 58)(56 57)(61 68)(62 67)(63 66)(64 65)(69 70)(71 78)(72 77)(73 76)(74 75)(79 80)(81 82)(83 90)(84 89)(85 88)(86 87)(91 98)(92 97)(93 96)(94 95)(99 100)(101 108)(102 107)(103 106)(104 105)(109 110)(111 112)(113 120)(114 119)(115 118)(116 117)
(1 16 29 117 40 105)(2 17 30 118 31 106)(3 18 21 119 32 107)(4 19 22 120 33 108)(5 20 23 111 34 109)(6 11 24 112 35 110)(7 12 25 113 36 101)(8 13 26 114 37 102)(9 14 27 115 38 103)(10 15 28 116 39 104)(41 83 66 71 53 96)(42 84 67 72 54 97)(43 85 68 73 55 98)(44 86 69 74 56 99)(45 87 70 75 57 100)(46 88 61 76 58 91)(47 89 62 77 59 92)(48 90 63 78 60 93)(49 81 64 79 51 94)(50 82 65 80 52 95)
(1 50 6 45)(2 43 5 42)(3 46 4 49)(7 48 10 47)(8 41 9 44)(11 95 16 100)(12 98 15 97)(13 91 14 94)(17 93 20 92)(18 96 19 99)(21 58 22 51)(23 54 30 55)(24 57 29 52)(25 60 28 59)(26 53 27 56)(31 68 34 67)(32 61 33 64)(35 70 40 65)(36 63 39 62)(37 66 38 69)(71 120 74 119)(72 113 73 116)(75 112 80 117)(76 115 79 114)(77 118 78 111)(81 102 88 103)(82 105 87 110)(83 108 86 107)(84 101 85 104)(89 106 90 109)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A ··· 4H 5 6A ··· 6G 6H ··· 6O 10A ··· 10G 15A 15B 30A ··· 30N order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 ··· 4 5 6 ··· 6 6 ··· 6 10 ··· 10 15 15 30 ··· 30 size 1 1 1 1 2 2 5 5 5 5 10 10 2 30 ··· 30 4 2 ··· 2 10 ··· 10 4 ··· 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + - + + + image C1 C2 C2 C2 C4 C4 S3 D4 Dic3 D6 Dic3 C3⋊D4 F5 C2×F5 C3⋊F5 C22⋊F5 C2×C3⋊F5 D10.D6 kernel C2×D10.D6 D10.D6 C22×C3⋊F5 D5×C22×C6 D5×C2×C6 C22×C30 C23×D5 C6×D5 C22×D5 C22×D5 C22×C10 D10 C22×C6 C2×C6 C23 C6 C22 C2 # reps 1 4 2 1 6 2 1 4 3 3 1 8 1 3 2 4 6 8

Matrix representation of C2×D10.D6 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 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
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 60 0 0 0 0 1 0 60 0 0 0 1 0 0 0 0 60 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 0 0 0 60 0 1 0 0 60 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 17 54 0 7 0 0 0 10 54 7 0 0 7 54 10 0 0 0 7 0 54 17
,
 0 50 0 0 0 0 11 0 0 0 0 0 0 0 7 14 47 54 0 0 0 14 54 7 0 0 14 0 47 7 0 0 14 7 0 54

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,1,1,1,1,0,0,60,0,0,0,0,0,0,60,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,1,1,1,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,0,7,7,0,0,54,10,54,0,0,0,0,54,10,54,0,0,7,7,0,17],[0,11,0,0,0,0,50,0,0,0,0,0,0,0,7,0,14,14,0,0,14,14,0,7,0,0,47,54,47,0,0,0,54,7,7,54] >;

C2×D10.D6 in GAP, Magma, Sage, TeX

C_2\times D_{10}.D_6
% in TeX

G:=Group("C2xD10.D6");
// GroupNames label

G:=SmallGroup(480,1072);
// by ID

G=gap.SmallGroup(480,1072);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,2693,14118,2379]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^6=1,e^2=b^4*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^7,c*d=d*c,e*c*e^-1=b^6*c,e*d*e^-1=b^5*d^-1>;
// generators/relations

׿
×
𝔽