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## G = C62.81D6order 432 = 24·33

### 29th non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.81D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — Dic3×C3×C6 — C62.81D6
 Lower central C33 — C32×C6 — C62.81D6
 Upper central C1 — C22

Generators and relations for C62.81D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=b3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 984 in 220 conjugacy classes, 72 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×C12, C62, C62, C62, Dic3⋊C4, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C335C4, C3×C62, C62.C22, C6.Dic6, Dic3×C3×C6, C6×C3⋊Dic3, C2×C335C4, C62.81D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, C3⋊S3, Dic6, C4×S3, C3⋊D4, S32, C2×C3⋊S3, Dic3⋊C4, C6.D6, D6⋊S3, C322Q8, C324Q8, C4×C3⋊S3, C327D4, S3×C3⋊S3, C62.C22, C6.Dic6, C338(C2×C4), C336D4, C334Q8, C62.81D6

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

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

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

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

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 4 4 4 4 type + + + + + + + - + - + + - - image C1 C2 C2 C2 C4 S3 S3 D4 Q8 D6 Dic6 C4×S3 C3⋊D4 S32 C6.D6 D6⋊S3 C32⋊2Q8 kernel C62.81D6 Dic3×C3×C6 C6×C3⋊Dic3 C2×C33⋊5C4 C33⋊5C4 C6×Dic3 C2×C3⋊Dic3 C32×C6 C32×C6 C62 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 # reps 1 1 1 1 4 4 1 1 1 5 10 10 10 4 4 4 4

Matrix representation of C62.81D6 in GL8(𝔽13)

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

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

C62.81D6 in GAP, Magma, Sage, TeX

`C_6^2._{81}D_6`
`% in TeX`

`G:=Group("C6^2.81D6");`
`// GroupNames label`

`G:=SmallGroup(432,453);`
`// by ID`

`G=gap.SmallGroup(432,453);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,36,571,2028,14118]);`
`// Polycyclic`

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