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## G = C62.30D6order 432 = 24·33

### 13rd 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 — C62.30D6
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C2×He3⋊3C4 — C62.30D6
 Lower central He3 — C2×He3 — C62.30D6
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C62.30D6
G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, cac-1=ab2, dad-1=a-1b4, bc=cb, bd=db, dcd-1=b3c5 >

Subgroups: 453 in 143 conjugacy classes, 59 normal (19 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C2×He3, C6×Dic3, C6×C12, He33C4, C4×He3, C22×He3, C3×C4⋊Dic3, C2×He33C4, C2×C4×He3, C62.30D6
Quotients:

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

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

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

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

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 Dic3 D6 Dic6 D12 He3⋊C2 He3⋊3C4 C2×He3⋊C2 He3⋊4Q8 He3⋊5D4 kernel C62.30D6 C2×He3⋊3C4 C2×C4×He3 C4×He3 C6×C12 C2×He3 C2×He3 C3×C12 C62 C3×C6 C3×C6 C2×C4 C4 C22 C2 C2 # reps 1 2 1 4 4 1 1 8 4 8 8 4 8 4 2 2

Matrix representation of C62.30D6 in GL7(𝔽13)

 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
,
 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 9
,
 6 3 0 0 0 0 0 10 3 0 0 0 0 0 0 0 3 10 0 0 0 0 0 3 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 9
,
 7 10 0 0 0 0 0 3 6 0 0 0 0 0 0 0 2 11 0 0 0 0 0 9 11 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 4 0 0 0 0 0 10 0

`G:=sub<GL(7,GF(13))| [0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9],[6,10,0,0,0,0,0,3,3,0,0,0,0,0,0,0,3,3,0,0,0,0,0,10,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9],[7,3,0,0,0,0,0,10,6,0,0,0,0,0,0,0,2,9,0,0,0,0,0,11,11,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,0,0,0,0,0,4,0] >;`

C62.30D6 in GAP, Magma, Sage, TeX

`C_6^2._{30}D_6`
`% in TeX`

`G:=Group("C6^2.30D6");`
`// GroupNames label`

`G:=SmallGroup(432,188);`
`// by ID`

`G=gap.SmallGroup(432,188);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,64,1124,4037,537]);`
`// Polycyclic`

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

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