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## G = C62.113D4order 288 = 25·32

### 18th non-split extension by C62 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.113D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C12 — C2×C12⋊S3 — C62.113D4
 Lower central C32 — C3×C6 — C3×C12 — C62.113D4
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C62.113D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=a3b3c3 >

Subgroups: 772 in 150 conjugacy classes, 59 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4, C22, C22 [×4], S3 [×8], C6 [×12], C8, C2×C4, C2×C4, D4 [×3], C23, C32, C12 [×8], C12 [×4], D6 [×16], C2×C6 [×4], C4⋊C4, C2×C8, C2×D4, C3⋊S3 [×2], C3×C6 [×3], C3⋊C8 [×4], D12 [×12], C2×C12 [×4], C2×C12 [×4], C22×S3 [×4], D4⋊C4, C3×C12 [×2], C3×C12, C2×C3⋊S3 [×4], C62, C2×C3⋊C8 [×4], C3×C4⋊C4 [×4], C2×D12 [×4], C324C8, C12⋊S3 [×2], C12⋊S3, C6×C12, C6×C12, C22×C3⋊S3, C6.D8 [×4], C2×C324C8, C32×C4⋊C4, C2×C12⋊S3, C62.113D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, D8, SD16, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], D4⋊C4, C2×C3⋊S3, D6⋊C4 [×4], D4⋊S3 [×4], Q82S3 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C6.D8 [×4], C6.11D12, C327D8, C3211SD16, C62.113D4

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 8A 8B 8C 8D 12A ··· 12X order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 36 36 2 2 2 2 2 2 4 4 2 ··· 2 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 D8 SD16 C4×S3 D12 C3⋊D4 D4⋊S3 Q8⋊2S3 kernel C62.113D4 C2×C32⋊4C8 C32×C4⋊C4 C2×C12⋊S3 C12⋊S3 C3×C4⋊C4 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C12 C2×C6 C6 C6 # reps 1 1 1 1 4 4 1 1 4 2 2 8 8 8 4 4

Matrix representation of C62.113D4 in GL6(𝔽73)

 0 72 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 7 14 0 0 0 0 7 66 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 16 57 0 0 0 0 16 16
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[7,7,0,0,0,0,14,66,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;`

C62.113D4 in GAP, Magma, Sage, TeX

`C_6^2._{113}D_4`
`% in TeX`

`G:=Group("C6^2.113D4");`
`// GroupNames label`

`G:=SmallGroup(288,284);`
`// by ID`

`G=gap.SmallGroup(288,284);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,675,346,80,2693,9414]);`
`// Polycyclic`

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

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