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## G = C62.82D6order 432 = 24·33

### 30th non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.82D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — Dic3×C3×C6 — C62.82D6
 Lower central C33 — C32×C6 — C62.82D6
 Upper central C1 — C22

Generators and relations for C62.82D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 984 in 220 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C335C4, C3×C62, Dic3⋊Dic3, C6.Dic6, Dic3×C3×C6, C6×C3⋊Dic3, C2×C335C4, C62.82D6
Quotients:

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 4F 6A ··· 6O 6P ··· 6AA 12A ··· 12P 12Q 12R 12S 12T 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 size 1 1 1 1 2 ··· 2 4 4 4 4 6 6 18 18 54 54 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - - + - + + - + - image C1 C2 C2 C2 C4 S3 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 C3⋊D12 C32⋊2Q8 kernel C62.82D6 Dic3×C3×C6 C6×C3⋊Dic3 C2×C33⋊5C4 C3×C3⋊Dic3 C6×Dic3 C2×C3⋊Dic3 C32×C6 C32×C6 C3⋊Dic3 C62 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 # reps 1 1 1 1 4 4 1 1 1 2 5 10 8 2 8 4 4 4 4

Matrix representation of C62.82D6 in GL6(𝔽13)

 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 7 3 0 0 0 0 10 10 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 5 8 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 8 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12

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

C62.82D6 in GAP, Magma, Sage, TeX

`C_6^2._{82}D_6`
`% in TeX`

`G:=Group("C6^2.82D6");`
`// GroupNames label`

`G:=SmallGroup(432,454);`
`// by ID`

`G=gap.SmallGroup(432,454);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,64,571,2028,14118]);`
`// Polycyclic`

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

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