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## G = C62.258C23order 288 = 25·32

### 103rd non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.258C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C12⋊S3 — C62.258C23
 Lower central C32 — C62 — C62.258C23
 Upper central C1 — C22 — C2×D4

Generators and relations for C62.258C23
G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=b3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ece=a3c, ede=b3d >

Subgroups: 1420 in 324 conjugacy classes, 81 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C4 [×4], C22, C22 [×12], S3 [×8], C6 [×12], C6 [×8], C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], C32, Dic3 [×16], C12 [×8], D6 [×24], C2×C6 [×4], C2×C6 [×24], C42, C2×D4, C2×D4 [×5], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], D12 [×8], C2×Dic3 [×8], C3⋊D4 [×32], C2×C12 [×4], C3×D4 [×8], C22×S3 [×8], C22×C6 [×8], C41D4, C3⋊Dic3 [×4], C3×C12 [×2], C2×C3⋊S3 [×6], C62, C62 [×6], C4×Dic3 [×4], C2×D12 [×4], C2×C3⋊D4 [×16], C6×D4 [×4], C12⋊S3 [×2], C2×C3⋊Dic3 [×2], C327D4 [×8], C6×C12, D4×C32 [×2], C22×C3⋊S3 [×2], C2×C62 [×2], C123D4 [×4], C4×C3⋊Dic3, C2×C12⋊S3, C2×C327D4 [×4], D4×C3×C6, C62.258C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×6], C23, D6 [×12], C2×D4 [×3], C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C41D4, C2×C3⋊S3 [×3], S3×D4 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C123D4 [×4], D4×C3⋊S3 [×2], C2×C327D4, C62.258C23

Smallest permutation representation of C62.258C23
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 15 58 33 39 61)(2 16 59 34 40 62)(3 17 60 35 41 63)(4 18 55 36 42 64)(5 13 56 31 37 65)(6 14 57 32 38 66)(7 22 123 116 29 140)(8 23 124 117 30 141)(9 24 125 118 25 142)(10 19 126 119 26 143)(11 20 121 120 27 144)(12 21 122 115 28 139)(43 92 67 73 101 50)(44 93 68 74 102 51)(45 94 69 75 97 52)(46 95 70 76 98 53)(47 96 71 77 99 54)(48 91 72 78 100 49)(79 128 103 109 137 86)(80 129 104 110 138 87)(81 130 105 111 133 88)(82 131 106 112 134 89)(83 132 107 113 135 90)(84 127 108 114 136 85)
(2 6)(3 5)(7 21)(8 20)(9 19)(10 24)(11 23)(12 22)(13 63)(14 62)(15 61)(16 66)(17 65)(18 64)(25 119)(26 118)(27 117)(28 116)(29 115)(30 120)(31 35)(32 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 52)(50 51)(53 54)(67 68)(69 72)(70 71)(73 93)(74 92)(75 91)(76 96)(77 95)(78 94)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(109 135)(110 134)(111 133)(112 138)(113 137)(114 136)(121 141)(122 140)(123 139)(124 144)(125 143)(126 142)
(1 105 33 88)(2 106 34 89)(3 107 35 90)(4 108 36 85)(5 103 31 86)(6 104 32 87)(7 74 116 44)(8 75 117 45)(9 76 118 46)(10 77 119 47)(11 78 120 48)(12 73 115 43)(13 109 37 79)(14 110 38 80)(15 111 39 81)(16 112 40 82)(17 113 41 83)(18 114 42 84)(19 99 26 96)(20 100 27 91)(21 101 28 92)(22 102 29 93)(23 97 30 94)(24 98 25 95)(49 144 72 121)(50 139 67 122)(51 140 68 123)(52 141 69 124)(53 142 70 125)(54 143 71 126)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 133 61 130)(59 134 62 131)(60 135 63 132)
(1 52)(2 53)(3 54)(4 49)(5 50)(6 51)(7 80)(8 81)(9 82)(10 83)(11 84)(12 79)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 132)(20 127)(21 128)(22 129)(23 130)(24 131)(25 134)(26 135)(27 136)(28 137)(29 138)(30 133)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(85 144)(86 139)(87 140)(88 141)(89 142)(90 143)(103 122)(104 123)(105 124)(106 125)(107 126)(108 121)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 S3×D4 kernel C62.258C23 C4×C3⋊Dic3 C2×C12⋊S3 C2×C32⋊7D4 D4×C3×C6 C6×D4 C3⋊Dic3 C3×C12 C2×C12 C22×C6 C12 C6 # reps 1 1 1 4 1 4 4 2 4 8 16 8

Matrix representation of C62.258C23 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 8 12
,
 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 1 3 0 0 0 0 8 12
,
 2 9 0 0 0 0 4 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 5 1

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

C62.258C23 in GAP, Magma, Sage, TeX

`C_6^2._{258}C_2^3`
`% in TeX`

`G:=Group("C6^2.258C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,797);`
`// by ID`

`G=gap.SmallGroup(288,797);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,219,2693,9414]);`
`// Polycyclic`

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

׿
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