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## G = C62.20D6order 432 = 24·33

### 3rd non-split extension by C62 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.20D6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×He3 — C2×C32⋊C12 — C62.20D6
 Lower central C32 — C3×C6 — C62.20D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.20D6
G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 397 in 107 conjugacy classes, 46 normal (32 characteristic)
C1, C2 [×3], C3, C3 [×3], C4 [×2], C4 [×2], C22, C6 [×3], C6 [×9], C2×C4, C2×C4 [×2], C32 [×2], C32, Dic3 [×4], C12 [×2], C12 [×8], C2×C6, C2×C6 [×3], C4⋊C4, C3×C6 [×6], C3×C6 [×3], C2×Dic3 [×4], C2×C12, C2×C12 [×5], He3, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×4], C3×C12 [×2], C62 [×2], C62, C4⋊Dic3 [×2], C3×C4⋊C4, C2×He3 [×3], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12 [×2], C6×C12, C32⋊C12 [×2], C4×He3 [×2], C22×He3, C3×C4⋊Dic3, C12⋊Dic3, C2×C32⋊C12 [×2], C2×C4×He3, C62.20D6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3 [×2], S3×C6, C4⋊Dic3, C3×C4⋊C4, C32⋊C6, C3×Dic6, C3×D12, C6×Dic3, C32⋊C12 [×2], C2×C32⋊C6, C3×C4⋊Dic3, He33Q8, He34D4, C2×C32⋊C12, C62.20D6

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 type + + + + + - - + - + + - + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 Dic3 D6 C3×S3 Dic6 D12 C3×D4 C3×Q8 C3×Dic3 S3×C6 C3×Dic6 C3×D12 C32⋊C6 C32⋊C12 C2×C32⋊C6 He3⋊3Q8 He3⋊4D4 kernel C62.20D6 C2×C32⋊C12 C2×C4×He3 C12⋊Dic3 C4×He3 C2×C3⋊Dic3 C6×C12 C3×C12 C6×C12 C2×He3 C2×He3 C3×C12 C62 C2×C12 C3×C6 C3×C6 C3×C6 C3×C6 C12 C2×C6 C6 C6 C2×C4 C4 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 1 2 1 2 2 2 2 2 4 2 4 4 1 2 1 2 2

Matrix representation of C62.20D6 in GL8(𝔽13)

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

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

C62.20D6 in GAP, Magma, Sage, TeX

`C_6^2._{20}D_6`
`% in TeX`

`G:=Group("C6^2.20D6");`
`// GroupNames label`

`G:=SmallGroup(432,140);`
`// by ID`

`G=gap.SmallGroup(432,140);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,4037,2035,14118]);`
`// 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=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;`
`// generators/relations`

׿
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