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## G = C62.247C23order 288 = 25·32

### 92nd non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.247C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.247C23
 Lower central C32 — C3×C6 — C62.247C23
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C62.247C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=a3, d2=b3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, de=ed >

Subgroups: 580 in 228 conjugacy classes, 125 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C32, Dic3 [×16], C12 [×16], C2×C6 [×12], C2×C6 [×8], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×16], C2×C12 [×24], C22×C6 [×4], C42⋊C2, C3⋊Dic3 [×4], C3×C12 [×4], C62, C62 [×2], C62 [×2], C4×Dic3 [×8], C4⋊Dic3 [×8], C6.D4 [×8], C22×C12 [×4], C2×C3⋊Dic3 [×4], C6×C12 [×2], C6×C12 [×4], C2×C62, C23.26D6 [×4], C4×C3⋊Dic3 [×2], C12⋊Dic3 [×2], C625C4 [×2], C2×C6×C12, C62.247C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, Dic3 [×16], D6 [×12], C22×C4, C4○D4 [×2], C3⋊S3, C2×Dic3 [×24], C22×S3 [×4], C42⋊C2, C3⋊Dic3 [×4], C2×C3⋊S3 [×3], C4○D12 [×8], C22×Dic3 [×4], C2×C3⋊Dic3 [×6], C22×C3⋊S3, C23.26D6 [×4], C12.59D6 [×2], C22×C3⋊Dic3, C62.247C23

Smallest permutation representation of C62.247C23
On 144 points
Generators in S144
```(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 32 58 16 39 61)(2 33 59 17 40 62)(3 34 60 18 41 63)(4 35 55 13 42 64)(5 36 56 14 37 65)(6 31 57 15 38 66)(7 29 138 125 24 142)(8 30 133 126 19 143)(9 25 134 121 20 144)(10 26 135 122 21 139)(11 27 136 123 22 140)(12 28 137 124 23 141)(43 71 77 53 94 82)(44 72 78 54 95 83)(45 67 73 49 96 84)(46 68 74 50 91 79)(47 69 75 51 92 80)(48 70 76 52 93 81)(85 118 108 128 111 98)(86 119 103 129 112 99)(87 120 104 130 113 100)(88 115 105 131 114 101)(89 116 106 132 109 102)(90 117 107 127 110 97)
(1 89 4 86)(2 88 5 85)(3 87 6 90)(7 53 10 50)(8 52 11 49)(9 51 12 54)(13 129 16 132)(14 128 17 131)(15 127 18 130)(19 81 22 84)(20 80 23 83)(21 79 24 82)(25 75 28 78)(26 74 29 77)(27 73 30 76)(31 97 34 100)(32 102 35 99)(33 101 36 98)(37 108 40 105)(38 107 41 104)(39 106 42 103)(43 122 46 125)(44 121 47 124)(45 126 48 123)(55 112 58 109)(56 111 59 114)(57 110 60 113)(61 116 64 119)(62 115 65 118)(63 120 66 117)(67 133 70 136)(68 138 71 135)(69 137 72 134)(91 142 94 139)(92 141 95 144)(93 140 96 143)
(1 50 16 46)(2 51 17 47)(3 52 18 48)(4 53 13 43)(5 54 14 44)(6 49 15 45)(7 132 125 89)(8 127 126 90)(9 128 121 85)(10 129 122 86)(11 130 123 87)(12 131 124 88)(19 117 30 110)(20 118 25 111)(21 119 26 112)(22 120 27 113)(23 115 28 114)(24 116 29 109)(31 96 38 67)(32 91 39 68)(33 92 40 69)(34 93 41 70)(35 94 42 71)(36 95 37 72)(55 82 64 77)(56 83 65 78)(57 84 66 73)(58 79 61 74)(59 80 62 75)(60 81 63 76)(97 143 107 133)(98 144 108 134)(99 139 103 135)(100 140 104 136)(101 141 105 137)(102 142 106 138)
(7 125)(8 126)(9 121)(10 122)(11 123)(12 124)(19 30)(20 25)(21 26)(22 27)(23 28)(24 29)(85 128)(86 129)(87 130)(88 131)(89 132)(90 127)(97 107)(98 108)(99 103)(100 104)(101 105)(102 106)(109 116)(110 117)(111 118)(112 119)(113 120)(114 115)(133 143)(134 144)(135 139)(136 140)(137 141)(138 142)```

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G ··· 4N 6A ··· 6AB 12A ··· 12AF order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 18 ··· 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - + + image C1 C2 C2 C2 C2 C4 S3 Dic3 D6 D6 C4○D4 C4○D12 kernel C62.247C23 C4×C3⋊Dic3 C12⋊Dic3 C62⋊5C4 C2×C6×C12 C6×C12 C22×C12 C2×C12 C2×C12 C22×C6 C3×C6 C6 # reps 1 2 2 2 1 8 4 16 8 4 4 32

Matrix representation of C62.247C23 in GL6(𝔽13)

 3 2 0 0 0 0 0 9 0 0 0 0 0 0 4 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 3 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 8 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,2,9,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,3,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C62.247C23 in GAP, Magma, Sage, TeX

`C_6^2._{247}C_2^3`
`% in TeX`

`G:=Group("C6^2.247C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,783);`
`// by ID`

`G=gap.SmallGroup(288,783);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,422,2693,9414]);`
`// Polycyclic`

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

׿
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