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## G = C62.78D6order 432 = 24·33

### 26th non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.78D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — Dic3×C3×C6 — C62.78D6
 Lower central C33 — C32×C6 — C62.78D6
 Upper central C1 — C22

Generators and relations for C62.78D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 1336 in 268 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C6.D4, C3×C3⋊S3, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C32×Dic3, C335C4, C6×C3⋊S3, C6×C3⋊S3, C3×C62, D6⋊Dic3, C6.11D12, Dic3×C3×C6, C2×C335C4, C2×C6×C3⋊S3, C62.78D6
Quotients:

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 6A ··· 6O 6P ··· 6AA 6AB 6AC 6AD 6AE 12A ··· 12P order 1 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 1 1 18 18 2 ··· 2 4 4 4 4 6 6 54 54 2 ··· 2 4 ··· 4 18 18 18 18 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + - - + image C1 C2 C2 C2 C4 S3 S3 D4 Dic3 D6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 D6⋊S3 C3⋊D12 kernel C62.78D6 Dic3×C3×C6 C2×C33⋊5C4 C2×C6×C3⋊S3 C6×C3⋊S3 C6×Dic3 C22×C3⋊S3 C32×C6 C2×C3⋊S3 C62 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 # reps 1 1 1 1 4 4 1 2 2 5 8 8 12 4 4 4 4

Matrix representation of C62.78D6 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 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 12 0
,
 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

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

C62.78D6 in GAP, Magma, Sage, TeX

`C_6^2._{78}D_6`
`% in TeX`

`G:=Group("C6^2.78D6");`
`// GroupNames label`

`G:=SmallGroup(432,450);`
`// by ID`

`G=gap.SmallGroup(432,450);`
`# by ID`

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

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,d^2=a^3*b^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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