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## G = C62.238C23order 288 = 25·32

### 83rd non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.238C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C62.238C23
 Lower central C32 — C62 — C62.238C23
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C62.238C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=a3, e2=b3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3c, ede-1=b3d >

Subgroups: 1036 in 234 conjugacy classes, 67 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C3 [×4], C4 [×5], C22, C22 [×7], S3 [×12], C6 [×12], C2×C4 [×3], C2×C4 [×4], D4 [×2], C23 [×2], C32, Dic3 [×8], C12 [×12], D6 [×28], C2×C6 [×4], C22⋊C4 [×3], C4⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3 [×3], C3×C6 [×3], C4×S3 [×8], D12 [×8], C2×Dic3 [×8], C2×C12 [×12], C22×S3 [×8], C22.D4, C3⋊Dic3 [×2], C3×C12 [×3], C2×C3⋊S3 [×2], C2×C3⋊S3 [×5], C62, Dic3⋊C4 [×4], D6⋊C4 [×12], C3×C4⋊C4 [×4], S3×C2×C4 [×4], C2×D12 [×4], C4×C3⋊S3 [×2], C12⋊S3 [×2], C2×C3⋊Dic3 [×2], C6×C12 [×3], C22×C3⋊S3 [×2], D6.D4 [×4], C6.Dic6, C6.11D12 [×3], C32×C4⋊C4, C2×C4×C3⋊S3, C2×C12⋊S3, C62.238C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C4○D4 [×2], C3⋊S3, C22×S3 [×4], C22.D4, C2×C3⋊S3 [×3], C4○D12 [×4], S3×D4 [×4], Q83S3 [×4], C22×C3⋊S3, D6.D4 [×4], C12.59D6, D4×C3⋊S3, C12.26D6, C62.238C23

Smallest permutation representation of C62.238C23
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 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 117 139 29 121)(8 23 118 140 30 122)(9 24 119 141 25 123)(10 19 120 142 26 124)(11 20 115 143 27 125)(12 21 116 144 28 126)(43 71 91 53 78 100)(44 72 92 54 73 101)(45 67 93 49 74 102)(46 68 94 50 75 97)(47 69 95 51 76 98)(48 70 96 52 77 99)(79 107 127 89 114 136)(80 108 128 90 109 137)(81 103 129 85 110 138)(82 104 130 86 111 133)(83 105 131 87 112 134)(84 106 132 88 113 135)
(1 118)(2 117)(3 116)(4 115)(5 120)(6 119)(7 59)(8 58)(9 57)(10 56)(11 55)(12 60)(13 121)(14 126)(15 125)(16 124)(17 123)(18 122)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 38)(26 37)(27 42)(28 41)(29 40)(30 39)(43 82)(44 81)(45 80)(46 79)(47 84)(48 83)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(61 140)(62 139)(63 144)(64 143)(65 142)(66 141)(67 137)(68 136)(69 135)(70 134)(71 133)(72 138)(73 129)(74 128)(75 127)(76 132)(77 131)(78 130)(91 111)(92 110)(93 109)(94 114)(95 113)(96 112)(97 107)(98 106)(99 105)(100 104)(101 103)(102 108)
(1 79 4 82)(2 80 5 83)(3 81 6 84)(7 74 10 77)(8 75 11 78)(9 76 12 73)(13 90 16 87)(14 85 17 88)(15 86 18 89)(19 99 22 102)(20 100 23 97)(21 101 24 98)(25 95 28 92)(26 96 29 93)(27 91 30 94)(31 106 34 103)(32 107 35 104)(33 108 36 105)(37 112 40 109)(38 113 41 110)(39 114 42 111)(43 118 46 115)(44 119 47 116)(45 120 48 117)(49 124 52 121)(50 125 53 122)(51 126 54 123)(55 130 58 127)(56 131 59 128)(57 132 60 129)(61 136 64 133)(62 137 65 134)(63 138 66 135)(67 142 70 139)(68 143 71 140)(69 144 72 141)
(1 50 18 46)(2 51 13 47)(3 52 14 48)(4 53 15 43)(5 54 16 44)(6 49 17 45)(7 103 139 110)(8 104 140 111)(9 105 141 112)(10 106 142 113)(11 107 143 114)(12 108 144 109)(19 132 26 135)(20 127 27 136)(21 128 28 137)(22 129 29 138)(23 130 30 133)(24 131 25 134)(31 74 38 67)(32 75 39 68)(33 76 40 69)(34 77 41 70)(35 78 42 71)(36 73 37 72)(55 100 64 91)(56 101 65 92)(57 102 66 93)(58 97 61 94)(59 98 62 95)(60 99 63 96)(79 115 89 125)(80 116 90 126)(81 117 85 121)(82 118 86 122)(83 119 87 123)(84 120 88 124)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 6A ··· 6L 12A ··· 12X order 1 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 18 18 36 2 2 2 2 2 2 4 4 18 18 36 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 C4○D4 C4○D12 S3×D4 Q8⋊3S3 kernel C62.238C23 C6.Dic6 C6.11D12 C32×C4⋊C4 C2×C4×C3⋊S3 C2×C12⋊S3 C3×C4⋊C4 C2×C3⋊S3 C2×C12 C3×C6 C6 C6 C6 # reps 1 1 3 1 1 1 4 2 12 4 16 4 4

Matrix representation of C62.238C23 in GL6(𝔽13)

 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 6 10 0 0 0 0 3 7
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 8 0 0 0 0 10 5 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 8 0 0 0 0 0 5 0 0 0 0 0 0 2 9 0 0 0 0 4 11

`G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,3,0,0,0,0,10,7],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,10,0,0,0,0,8,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,8,5,0,0,0,0,0,0,2,4,0,0,0,0,9,11] >;`

C62.238C23 in GAP, Magma, Sage, TeX

`C_6^2._{238}C_2^3`
`% in TeX`

`G:=Group("C6^2.238C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,751);`
`// by ID`

`G=gap.SmallGroup(288,751);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,100,2693,9414]);`
`// Polycyclic`

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

׿
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