Copied to
clipboard

## G = C62.D6order 432 = 24·33

### 2nd non-split extension by C62 of D6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C62.D6
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C2×C32⋊C12 — C62.D6
 Lower central He3 — C2×He3 — C62.D6
 Upper central C1 — C22

Generators and relations for C62.D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, cac-1=dad-1=a-1b4, cbc-1=b-1, bd=db, dcd-1=b3c5 >

Subgroups: 495 in 107 conjugacy classes, 35 normal (31 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C62, C62, Dic3⋊C4, C4⋊Dic3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C32⋊C12, C32⋊C12, He33C4, C22×He3, Dic3⋊Dic3, C62.C22, C2×C32⋊C12, C2×He33C4, C62.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, S32, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C322Q8, C32⋊D6, Dic3⋊Dic3, He32Q8, C6.S32, He33D4, C62.D6

Smallest permutation representation of C62.D6
On 144 points
Generators in S144
```(1 75 111 3 81 117)(2 114 78 4 120 84)(5 132 102 7 126 108)(6 105 123 8 99 129)(9 95 50 11 89 56)(10 53 86 12 59 92)(13 128 106 15 122 100)(14 97 131 16 103 125)(17 136 69 19 142 63)(18 72 139 20 66 133)(21 79 119 23 73 113)(22 110 82 24 116 76)(25 70 141 27 64 135)(26 144 61 28 138 67)(29 68 143 31 62 137)(30 134 71 32 140 65)(33 130 104 35 124 98)(34 107 121 36 101 127)(37 55 96 39 49 90)(38 87 58 40 93 52)(41 112 80 43 118 74)(42 83 115 44 77 109)(45 51 88 47 57 94)(46 91 54 48 85 60)
(1 6 42 36 21 14)(2 15 22 33 43 7)(3 8 44 34 23 16)(4 13 24 35 41 5)(9 26 46 32 38 17)(10 18 39 29 47 27)(11 28 48 30 40 19)(12 20 37 31 45 25)(49 68 57 64 53 72)(50 61 54 65 58 69)(51 70 59 66 55 62)(52 63 56 67 60 71)(73 103 81 99 77 107)(74 108 78 100 82 104)(75 105 83 101 79 97)(76 98 80 102 84 106)(85 134 93 142 89 138)(86 139 90 143 94 135)(87 136 95 144 91 140)(88 141 92 133 96 137)(109 121 113 125 117 129)(110 130 118 126 114 122)(111 123 115 127 119 131)(112 132 120 128 116 124)
(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)
(1 10 3 12)(2 30 4 32)(5 46 7 48)(6 18 8 20)(9 33 11 35)(13 38 15 40)(14 27 16 25)(17 22 19 24)(21 47 23 45)(26 43 28 41)(29 34 31 36)(37 42 39 44)(49 111 55 117)(50 132 56 126)(51 109 57 115)(52 130 58 124)(53 119 59 113)(54 128 60 122)(61 120 67 114)(62 129 68 123)(63 118 69 112)(64 127 70 121)(65 116 71 110)(66 125 72 131)(73 90 79 96)(74 140 80 134)(75 88 81 94)(76 138 82 144)(77 86 83 92)(78 136 84 142)(85 104 91 98)(87 102 93 108)(89 100 95 106)(97 137 103 143)(99 135 105 141)(101 133 107 139)```

`G:=sub<Sym(144)| (1,75,111,3,81,117)(2,114,78,4,120,84)(5,132,102,7,126,108)(6,105,123,8,99,129)(9,95,50,11,89,56)(10,53,86,12,59,92)(13,128,106,15,122,100)(14,97,131,16,103,125)(17,136,69,19,142,63)(18,72,139,20,66,133)(21,79,119,23,73,113)(22,110,82,24,116,76)(25,70,141,27,64,135)(26,144,61,28,138,67)(29,68,143,31,62,137)(30,134,71,32,140,65)(33,130,104,35,124,98)(34,107,121,36,101,127)(37,55,96,39,49,90)(38,87,58,40,93,52)(41,112,80,43,118,74)(42,83,115,44,77,109)(45,51,88,47,57,94)(46,91,54,48,85,60), (1,6,42,36,21,14)(2,15,22,33,43,7)(3,8,44,34,23,16)(4,13,24,35,41,5)(9,26,46,32,38,17)(10,18,39,29,47,27)(11,28,48,30,40,19)(12,20,37,31,45,25)(49,68,57,64,53,72)(50,61,54,65,58,69)(51,70,59,66,55,62)(52,63,56,67,60,71)(73,103,81,99,77,107)(74,108,78,100,82,104)(75,105,83,101,79,97)(76,98,80,102,84,106)(85,134,93,142,89,138)(86,139,90,143,94,135)(87,136,95,144,91,140)(88,141,92,133,96,137)(109,121,113,125,117,129)(110,130,118,126,114,122)(111,123,115,127,119,131)(112,132,120,128,116,124), (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), (1,10,3,12)(2,30,4,32)(5,46,7,48)(6,18,8,20)(9,33,11,35)(13,38,15,40)(14,27,16,25)(17,22,19,24)(21,47,23,45)(26,43,28,41)(29,34,31,36)(37,42,39,44)(49,111,55,117)(50,132,56,126)(51,109,57,115)(52,130,58,124)(53,119,59,113)(54,128,60,122)(61,120,67,114)(62,129,68,123)(63,118,69,112)(64,127,70,121)(65,116,71,110)(66,125,72,131)(73,90,79,96)(74,140,80,134)(75,88,81,94)(76,138,82,144)(77,86,83,92)(78,136,84,142)(85,104,91,98)(87,102,93,108)(89,100,95,106)(97,137,103,143)(99,135,105,141)(101,133,107,139)>;`

`G:=Group( (1,75,111,3,81,117)(2,114,78,4,120,84)(5,132,102,7,126,108)(6,105,123,8,99,129)(9,95,50,11,89,56)(10,53,86,12,59,92)(13,128,106,15,122,100)(14,97,131,16,103,125)(17,136,69,19,142,63)(18,72,139,20,66,133)(21,79,119,23,73,113)(22,110,82,24,116,76)(25,70,141,27,64,135)(26,144,61,28,138,67)(29,68,143,31,62,137)(30,134,71,32,140,65)(33,130,104,35,124,98)(34,107,121,36,101,127)(37,55,96,39,49,90)(38,87,58,40,93,52)(41,112,80,43,118,74)(42,83,115,44,77,109)(45,51,88,47,57,94)(46,91,54,48,85,60), (1,6,42,36,21,14)(2,15,22,33,43,7)(3,8,44,34,23,16)(4,13,24,35,41,5)(9,26,46,32,38,17)(10,18,39,29,47,27)(11,28,48,30,40,19)(12,20,37,31,45,25)(49,68,57,64,53,72)(50,61,54,65,58,69)(51,70,59,66,55,62)(52,63,56,67,60,71)(73,103,81,99,77,107)(74,108,78,100,82,104)(75,105,83,101,79,97)(76,98,80,102,84,106)(85,134,93,142,89,138)(86,139,90,143,94,135)(87,136,95,144,91,140)(88,141,92,133,96,137)(109,121,113,125,117,129)(110,130,118,126,114,122)(111,123,115,127,119,131)(112,132,120,128,116,124), (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), (1,10,3,12)(2,30,4,32)(5,46,7,48)(6,18,8,20)(9,33,11,35)(13,38,15,40)(14,27,16,25)(17,22,19,24)(21,47,23,45)(26,43,28,41)(29,34,31,36)(37,42,39,44)(49,111,55,117)(50,132,56,126)(51,109,57,115)(52,130,58,124)(53,119,59,113)(54,128,60,122)(61,120,67,114)(62,129,68,123)(63,118,69,112)(64,127,70,121)(65,116,71,110)(66,125,72,131)(73,90,79,96)(74,140,80,134)(75,88,81,94)(76,138,82,144)(77,86,83,92)(78,136,84,142)(85,104,91,98)(87,102,93,108)(89,100,95,106)(97,137,103,143)(99,135,105,141)(101,133,107,139) );`

`G=PermutationGroup([[(1,75,111,3,81,117),(2,114,78,4,120,84),(5,132,102,7,126,108),(6,105,123,8,99,129),(9,95,50,11,89,56),(10,53,86,12,59,92),(13,128,106,15,122,100),(14,97,131,16,103,125),(17,136,69,19,142,63),(18,72,139,20,66,133),(21,79,119,23,73,113),(22,110,82,24,116,76),(25,70,141,27,64,135),(26,144,61,28,138,67),(29,68,143,31,62,137),(30,134,71,32,140,65),(33,130,104,35,124,98),(34,107,121,36,101,127),(37,55,96,39,49,90),(38,87,58,40,93,52),(41,112,80,43,118,74),(42,83,115,44,77,109),(45,51,88,47,57,94),(46,91,54,48,85,60)], [(1,6,42,36,21,14),(2,15,22,33,43,7),(3,8,44,34,23,16),(4,13,24,35,41,5),(9,26,46,32,38,17),(10,18,39,29,47,27),(11,28,48,30,40,19),(12,20,37,31,45,25),(49,68,57,64,53,72),(50,61,54,65,58,69),(51,70,59,66,55,62),(52,63,56,67,60,71),(73,103,81,99,77,107),(74,108,78,100,82,104),(75,105,83,101,79,97),(76,98,80,102,84,106),(85,134,93,142,89,138),(86,139,90,143,94,135),(87,136,95,144,91,140),(88,141,92,133,96,137),(109,121,113,125,117,129),(110,130,118,126,114,122),(111,123,115,127,119,131),(112,132,120,128,116,124)], [(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)], [(1,10,3,12),(2,30,4,32),(5,46,7,48),(6,18,8,20),(9,33,11,35),(13,38,15,40),(14,27,16,25),(17,22,19,24),(21,47,23,45),(26,43,28,41),(29,34,31,36),(37,42,39,44),(49,111,55,117),(50,132,56,126),(51,109,57,115),(52,130,58,124),(53,119,59,113),(54,128,60,122),(61,120,67,114),(62,129,68,123),(63,118,69,112),(64,127,70,121),(65,116,71,110),(66,125,72,131),(73,90,79,96),(74,140,80,134),(75,88,81,94),(76,138,82,144),(77,86,83,92),(78,136,84,142),(85,104,91,98),(87,102,93,108),(89,100,95,106),(97,137,103,143),(99,135,105,141),(101,133,107,139)]])`

38 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A ··· 4F 6A 6B 6C 6D ··· 6I 6J 6K 6L 12A ··· 12L order 1 2 2 2 3 3 3 3 4 ··· 4 6 6 6 6 ··· 6 6 6 6 12 ··· 12 size 1 1 1 1 2 6 6 12 18 ··· 18 2 2 2 6 ··· 6 12 12 12 18 ··· 18

38 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 6 6 6 6 type + + + + + - - + - + + - + - + - image C1 C2 C2 C4 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 C3⋊D12 C32⋊2Q8 C32⋊D6 He3⋊2Q8 C6.S32 He3⋊3D4 kernel C62.D6 C2×C32⋊C12 C2×He3⋊3C4 C32⋊C12 C2×C3⋊Dic3 C2×He3 C2×He3 C3⋊Dic3 C62 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 4 2 1 1 2 2 4 2 2 2 1 1 1 1 2 2 2 2

Matrix representation of C62.D6 in GL10(𝔽13)

 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 1 0 0 0 0 0 0 0 0 12 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 12 0 0 0 0 12 12 0 0 2 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 1 0 0 0 0 0 0 0 12 0 1 0 0 0 0 0 0 0 0 12 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 1 1 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 1 1 0 0 0 0 0 1 0 0 12 0
,
 6 0 6 0 0 0 0 0 0 0 12 8 12 8 0 0 0 0 0 0 12 0 7 0 0 0 0 0 0 0 11 3 1 5 0 0 0 0 0 0 0 0 0 0 5 8 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 8 0 0 5 0 0 0 0 0 5 0 0 0 8 8 0 0 0 0 0 8 0 5 0 0 0 0 0 0 0 8 5 0 0 0
,
 8 10 5 3 0 0 0 0 0 0 0 5 0 8 0 0 0 0 0 0 0 0 5 3 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 3 7 0 0 0 0 0 0 0 0 6 10 0 0 0 0 0 0 0 0 0 7 0 0 6 3 0 0 0 0 3 10 0 0 10 3 0 0 0 0 3 10 10 3 0 0 0 0 0 0 6 0 10 7 0 0

`G:=sub<GL(10,GF(13))| [0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,2,1,1,1,1,0,0,0,0,12,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,12,0,0,0,0,0,1,1,0,1,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0],[6,12,12,11,0,0,0,0,0,0,0,8,0,3,0,0,0,0,0,0,6,12,7,1,0,0,0,0,0,0,0,8,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,8,8,8,0,8,8,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,0,8,0,0],[8,0,0,0,0,0,0,0,0,0,10,5,0,0,0,0,0,0,0,0,5,0,5,0,0,0,0,0,0,0,3,8,3,8,0,0,0,0,0,0,0,0,0,0,3,6,0,3,3,6,0,0,0,0,7,10,7,10,10,0,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,3,3,0,0] >;`

C62.D6 in GAP, Magma, Sage, TeX

`C_6^2.D_6`
`% in TeX`

`G:=Group("C6^2.D6");`
`// GroupNames label`

`G:=SmallGroup(432,95);`
`// by ID`

`G=gap.SmallGroup(432,95);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,571,4037,537,14118,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=a^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c^5>;`
`// generators/relations`

׿
×
𝔽