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G = C62.262C23order 288 = 25·32

107th non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.262C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C12⋊S3 — C62.262C23
 Lower central C32 — C62 — C62.262C23
 Upper central C1 — C22 — C2×Q8

Generators and relations for C62.262C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=e2=b3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ece-1=a3c, ede-1=b3d >

Subgroups: 972 in 228 conjugacy classes, 77 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×4], C22, C22 [×6], S3 [×8], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×8], C12 [×8], C12 [×8], D6 [×24], C2×C6 [×4], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3⋊S3 [×2], C3×C6, C3×C6 [×2], D12 [×8], C2×Dic3 [×8], C2×C12 [×12], C3×Q8 [×8], C22×S3 [×8], C4.4D4, C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, C4×Dic3 [×4], D6⋊C4 [×16], C2×D12 [×4], C6×Q8 [×4], C12⋊S3 [×2], C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], Q8×C32 [×2], C22×C3⋊S3 [×2], C12.23D4 [×4], C4×C3⋊Dic3, C6.11D12 [×4], C2×C12⋊S3, Q8×C3×C6, C62.262C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C4○D4 [×2], C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C4.4D4, C2×C3⋊S3 [×3], Q83S3 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C12.23D4 [×4], C12.26D6 [×2], C2×C327D4, C62.262C23

Smallest permutation representation of C62.262C23
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 18 58 31 39 61)(2 13 59 32 40 62)(3 14 60 33 41 63)(4 15 55 34 42 64)(5 16 56 35 37 65)(6 17 57 36 38 66)(7 126 27 24 140 118)(8 121 28 19 141 119)(9 122 29 20 142 120)(10 123 30 21 143 115)(11 124 25 22 144 116)(12 125 26 23 139 117)(43 92 51 73 101 71)(44 93 52 74 102 72)(45 94 53 75 97 67)(46 95 54 76 98 68)(47 96 49 77 99 69)(48 91 50 78 100 70)(79 128 87 109 137 107)(80 129 88 110 138 108)(81 130 89 111 133 103)(82 131 90 112 134 104)(83 132 85 113 135 105)(84 127 86 114 136 106)
(2 6)(3 5)(7 119)(8 118)(9 117)(10 116)(11 115)(12 120)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 27)(20 26)(21 25)(22 30)(23 29)(24 28)(32 36)(33 35)(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 54)(50 53)(51 52)(67 70)(68 69)(71 72)(73 93)(74 92)(75 91)(76 96)(77 95)(78 94)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)(85 107)(86 106)(87 105)(88 104)(89 103)(90 108)(109 135)(110 134)(111 133)(112 138)(113 137)(114 136)(121 140)(122 139)(123 144)(124 143)(125 142)(126 141)
(1 89 31 103)(2 90 32 104)(3 85 33 105)(4 86 34 106)(5 87 35 107)(6 88 36 108)(7 95 24 98)(8 96 19 99)(9 91 20 100)(10 92 21 101)(11 93 22 102)(12 94 23 97)(13 112 40 82)(14 113 41 83)(15 114 42 84)(16 109 37 79)(17 110 38 80)(18 111 39 81)(25 74 116 44)(26 75 117 45)(27 76 118 46)(28 77 119 47)(29 78 120 48)(30 73 115 43)(49 141 69 121)(50 142 70 122)(51 143 71 123)(52 144 72 124)(53 139 67 125)(54 140 68 126)(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 53 31 67)(2 54 32 68)(3 49 33 69)(4 50 34 70)(5 51 35 71)(6 52 36 72)(7 134 24 131)(8 135 19 132)(9 136 20 127)(10 137 21 128)(11 138 22 129)(12 133 23 130)(13 76 40 46)(14 77 41 47)(15 78 42 48)(16 73 37 43)(17 74 38 44)(18 75 39 45)(25 80 116 110)(26 81 117 111)(27 82 118 112)(28 83 119 113)(29 84 120 114)(30 79 115 109)(55 100 64 91)(56 101 65 92)(57 102 66 93)(58 97 61 94)(59 98 62 95)(60 99 63 96)(85 121 105 141)(86 122 106 142)(87 123 107 143)(88 124 108 144)(89 125 103 139)(90 126 104 140)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 C4○D4 C3⋊D4 Q8⋊3S3 kernel C62.262C23 C4×C3⋊Dic3 C6.11D12 C2×C12⋊S3 Q8×C3×C6 C6×Q8 C3×C12 C2×C12 C3×C6 C12 C6 # reps 1 1 4 1 1 4 2 12 4 16 8

Matrix representation of C62.262C23 in GL6(𝔽13)

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

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

C62.262C23 in GAP, Magma, Sage, TeX

`C_6^2._{262}C_2^3`
`% in TeX`

`G:=Group("C6^2.262C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,804);`
`// by ID`

`G=gap.SmallGroup(288,804);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,219,100,2693,9414]);`
`// Polycyclic`

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

׿
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