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G = C62.29D6order 432 = 24·33

12nd non-split extension by C62 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C62.29D6
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C2×He3⋊3C4 — C62.29D6
 Lower central He3 — C2×He3 — C62.29D6
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C62.29D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, cac-1=ab4, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b3c5 >

Subgroups: 453 in 143 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3×C12, C62, Dic3⋊C4, C3×C4⋊C4, C2×He3, C6×Dic3, C6×C12, He33C4, He33C4, C4×He3, C22×He3, C3×Dic3⋊C4, C2×He33C4, C2×C4×He3, C62.29D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, C3⋊S3, Dic6, C4×S3, C3⋊D4, C2×C3⋊S3, Dic3⋊C4, He3⋊C2, C324Q8, C4×C3⋊S3, C327D4, C2×He3⋊C2, C6.Dic6, He34Q8, C4×He3⋊C2, He37D4, C62.29D6

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6R 12A 12B 12C 12D 12E ··· 12T 12U ··· 12AB order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 6 6 6 6 2 2 18 18 18 18 1 ··· 1 6 ··· 6 2 2 2 2 6 ··· 6 18 ··· 18

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 3 3 3 6 6 type + + + + + - + - image C1 C2 C2 C4 S3 D4 Q8 D6 Dic6 C4×S3 C3⋊D4 He3⋊C2 C2×He3⋊C2 C4×He3⋊C2 He3⋊4Q8 He3⋊7D4 kernel C62.29D6 C2×He3⋊3C4 C2×C4×He3 He3⋊3C4 C6×C12 C2×He3 C2×He3 C62 C3×C6 C3×C6 C3×C6 C2×C4 C22 C2 C2 C2 # reps 1 2 1 4 4 1 1 4 8 8 8 4 4 8 2 2

Matrix representation of C62.29D6 in GL5(𝔽13)

 0 1 0 0 0 12 12 0 0 0 0 0 12 3 12 0 0 0 10 0 0 0 0 0 4
,
 12 0 0 0 0 0 12 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 4 2 0 0 0 11 2 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 12 5 5
,
 1 1 0 0 0 0 12 0 0 0 0 0 8 0 0 0 0 12 5 5 0 0 0 0 8

`G:=sub<GL(5,GF(13))| [0,12,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,3,10,0,0,0,12,0,4],[12,0,0,0,0,0,12,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[4,11,0,0,0,2,2,0,0,0,0,0,8,0,12,0,0,0,0,5,0,0,0,8,5],[1,0,0,0,0,1,12,0,0,0,0,0,8,12,0,0,0,0,5,0,0,0,0,5,8] >;`

C62.29D6 in GAP, Magma, Sage, TeX

`C_6^2._{29}D_6`
`% in TeX`

`G:=Group("C6^2.29D6");`
`// GroupNames label`

`G:=SmallGroup(432,187);`
`// by ID`

`G=gap.SmallGroup(432,187);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,1124,4037,537]);`
`// 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=a*b^4,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^3*c^5>;`
`// generators/relations`

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