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## G = C62.116D4order 288 = 25·32

### 21st non-split extension by C62 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.116D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C12 — C12⋊Dic3 — C62.116D4
 Lower central C32 — C3×C6 — C3×C12 — C62.116D4
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C62.116D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3b3c3 >

Subgroups: 452 in 150 conjugacy classes, 69 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4, C22, C22 [×4], C6 [×12], C6 [×8], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C32, Dic3 [×4], C12 [×8], C2×C6 [×4], C2×C6 [×16], C4⋊C4, C2×C8, C2×D4, C3×C6 [×3], C3×C6 [×2], C3⋊C8 [×4], C2×Dic3 [×4], C2×C12 [×4], C3×D4 [×8], C3×D4 [×4], C22×C6 [×4], D4⋊C4, C3⋊Dic3, C3×C12 [×2], C62, C62 [×4], C2×C3⋊C8 [×4], C4⋊Dic3 [×4], C6×D4 [×4], C324C8, C2×C3⋊Dic3, C6×C12, D4×C32 [×2], D4×C32, C2×C62, D4⋊Dic3 [×4], C2×C324C8, C12⋊Dic3, D4×C3×C6, C62.116D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], Dic3 [×8], D6 [×4], C22⋊C4, D8, SD16, C3⋊S3, C2×Dic3 [×4], C3⋊D4 [×8], D4⋊C4, C3⋊Dic3 [×2], C2×C3⋊S3, D4⋊S3 [×4], D4.S3 [×4], C6.D4 [×4], C2×C3⋊Dic3, C327D4 [×2], D4⋊Dic3 [×4], C327D8, C329SD16, C625C4, C62.116D4

Smallest permutation representation of C62.116D4
On 144 points
Generators in S144
```(1 128 141 26 108 113)(2 114 109 27 142 121)(3 122 143 28 110 115)(4 116 111 29 144 123)(5 124 137 30 112 117)(6 118 105 31 138 125)(7 126 139 32 106 119)(8 120 107 25 140 127)(9 100 41 65 78 129)(10 130 79 66 42 101)(11 102 43 67 80 131)(12 132 73 68 44 103)(13 104 45 69 74 133)(14 134 75 70 46 97)(15 98 47 71 76 135)(16 136 77 72 48 99)(17 89 81 59 50 34)(18 35 51 60 82 90)(19 91 83 61 52 36)(20 37 53 62 84 92)(21 93 85 63 54 38)(22 39 55 64 86 94)(23 95 87 57 56 40)(24 33 49 58 88 96)
(1 81 71 5 85 67)(2 68 86 6 72 82)(3 83 65 7 87 69)(4 70 88 8 66 84)(9 32 40 13 28 36)(10 37 29 14 33 25)(11 26 34 15 30 38)(12 39 31 16 35 27)(17 98 112 21 102 108)(18 109 103 22 105 99)(19 100 106 23 104 110)(20 111 97 24 107 101)(41 119 95 45 115 91)(42 92 116 46 96 120)(43 113 89 47 117 93)(44 94 118 48 90 114)(49 140 130 53 144 134)(50 135 137 54 131 141)(51 142 132 55 138 136)(52 129 139 56 133 143)(57 74 122 61 78 126)(58 127 79 62 123 75)(59 76 124 63 80 128)(60 121 73 64 125 77)
(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 25 26 8)(2 7 27 32)(3 31 28 6)(4 5 29 30)(9 68 65 12)(10 11 66 67)(13 72 69 16)(14 15 70 71)(17 58 59 24)(18 23 60 57)(19 64 61 22)(20 21 62 63)(33 34 88 81)(35 40 82 87)(36 86 83 39)(37 38 84 85)(41 132 129 44)(42 43 130 131)(45 136 133 48)(46 47 134 135)(49 50 96 89)(51 56 90 95)(52 94 91 55)(53 54 92 93)(73 78 103 100)(74 99 104 77)(75 76 97 98)(79 80 101 102)(105 110 125 122)(106 121 126 109)(107 108 127 128)(111 112 123 124)(113 140 141 120)(114 119 142 139)(115 138 143 118)(116 117 144 137)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 6M ··· 6AB 8A 8B 8C 8D 12A ··· 12H order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 4 4 2 2 2 2 2 2 36 36 2 ··· 2 4 ··· 4 18 18 18 18 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C4 S3 D4 D4 D6 Dic3 D8 SD16 C3⋊D4 C3⋊D4 D4⋊S3 D4.S3 kernel C62.116D4 C2×C32⋊4C8 C12⋊Dic3 D4×C3×C6 D4×C32 C6×D4 C3×C12 C62 C2×C12 C3×D4 C3×C6 C3×C6 C12 C2×C6 C6 C6 # reps 1 1 1 1 4 4 1 1 4 8 2 2 8 8 4 4

Matrix representation of C62.116D4 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 9 0 0 0 0 0 2 65 0 0 0 0 0 0 9 0 0 0 0 0 0 65
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 64
,
 12 4 0 0 0 0 55 0 0 0 0 0 0 0 21 4 0 0 0 0 72 52 0 0 0 0 0 0 0 45 0 0 0 0 13 0
,
 12 4 0 0 0 0 55 61 0 0 0 0 0 0 21 4 0 0 0 0 72 52 0 0 0 0 0 0 0 45 0 0 0 0 60 0

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,9,2,0,0,0,0,0,65,0,0,0,0,0,0,9,0,0,0,0,0,0,65],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[12,55,0,0,0,0,4,0,0,0,0,0,0,0,21,72,0,0,0,0,4,52,0,0,0,0,0,0,0,13,0,0,0,0,45,0],[12,55,0,0,0,0,4,61,0,0,0,0,0,0,21,72,0,0,0,0,4,52,0,0,0,0,0,0,0,60,0,0,0,0,45,0] >;`

C62.116D4 in GAP, Magma, Sage, TeX

`C_6^2._{116}D_4`
`% in TeX`

`G:=Group("C6^2.116D4");`
`// GroupNames label`

`G:=SmallGroup(288,307);`
`// by ID`

`G=gap.SmallGroup(288,307);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,675,346,80,2693,9414]);`
`// Polycyclic`

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

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