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## G = C62.256C23order 288 = 25·32

### 101st non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.256C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C62.256C23
 Lower central C32 — C62 — C62.256C23
 Upper central C1 — C22 — C2×D4

Generators and relations for C62.256C23
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, cd=dc, ece=a3c, ede=b3d >

Subgroups: 1068 in 282 conjugacy classes, 79 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×3], C22, C22 [×10], S3 [×8], C6 [×12], C6 [×8], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C32, Dic3 [×12], C12 [×8], D6 [×16], C2×C6 [×4], C2×C6 [×24], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C3⋊S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×12], C3⋊D4 [×16], C2×C12 [×4], C3×D4 [×8], C22×S3 [×4], C22×C6 [×8], C4⋊D4, C3⋊Dic3 [×3], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×6], C4⋊Dic3 [×4], C6.D4 [×8], S3×C2×C4 [×4], C2×C3⋊D4 [×8], C6×D4 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C327D4 [×4], C6×C12, D4×C32 [×2], C22×C3⋊S3, C2×C62 [×2], D63D4 [×4], C12⋊Dic3, C625C4 [×2], C2×C4×C3⋊S3, C2×C327D4 [×2], D4×C3×C6, C62.256C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×4], C23, D6 [×12], C2×D4 [×2], C4○D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C4⋊D4, C2×C3⋊S3 [×3], S3×D4 [×4], D42S3 [×4], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, D63D4 [×4], D4×C3⋊S3, C12.D6, C2×C327D4, C62.256C23

Smallest permutation representation of C62.256C23
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 35 38 57 13 52)(2 36 39 58 14 53)(3 31 40 59 15 54)(4 32 41 60 16 49)(5 33 42 55 17 50)(6 34 37 56 18 51)(7 134 20 142 120 27)(8 135 21 143 115 28)(9 136 22 144 116 29)(10 137 23 139 117 30)(11 138 24 140 118 25)(12 133 19 141 119 26)(43 78 91 63 86 71)(44 73 92 64 87 72)(45 74 93 65 88 67)(46 75 94 66 89 68)(47 76 95 61 90 69)(48 77 96 62 85 70)(79 114 127 99 122 107)(80 109 128 100 123 108)(81 110 129 101 124 103)(82 111 130 102 125 104)(83 112 131 97 126 105)(84 113 132 98 121 106)
(2 6)(3 5)(7 10)(8 9)(11 12)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 118)(20 117)(21 116)(22 115)(23 120)(24 119)(25 133)(26 138)(27 137)(28 136)(29 135)(30 134)(31 50)(32 49)(33 54)(34 53)(35 52)(36 51)(43 87)(44 86)(45 85)(46 90)(47 89)(48 88)(55 59)(56 58)(61 75)(62 74)(63 73)(64 78)(65 77)(66 76)(67 70)(68 69)(71 72)(79 126)(80 125)(81 124)(82 123)(83 122)(84 121)(91 92)(93 96)(94 95)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(104 108)(105 107)(127 131)(128 130)(139 142)(140 141)(143 144)
(1 129 57 103)(2 130 58 104)(3 131 59 105)(4 132 60 106)(5 127 55 107)(6 128 56 108)(7 96 142 70)(8 91 143 71)(9 92 144 72)(10 93 139 67)(11 94 140 68)(12 95 141 69)(13 81 35 101)(14 82 36 102)(15 83 31 97)(16 84 32 98)(17 79 33 99)(18 80 34 100)(19 90 26 76)(20 85 27 77)(21 86 28 78)(22 87 29 73)(23 88 30 74)(24 89 25 75)(37 123 51 109)(38 124 52 110)(39 125 53 111)(40 126 54 112)(41 121 49 113)(42 122 50 114)(43 135 63 115)(44 136 64 116)(45 137 65 117)(46 138 66 118)(47 133 61 119)(48 134 62 120)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 106)(8 107)(9 108)(10 103)(11 104)(12 105)(13 65)(14 66)(15 61)(16 62)(17 63)(18 64)(19 112)(20 113)(21 114)(22 109)(23 110)(24 111)(25 125)(26 126)(27 121)(28 122)(29 123)(30 124)(31 47)(32 48)(33 43)(34 44)(35 45)(36 46)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(79 135)(80 136)(81 137)(82 138)(83 133)(84 134)(97 119)(98 120)(99 115)(100 116)(101 117)(102 118)(127 143)(128 144)(129 139)(130 140)(131 141)(132 142)```

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

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

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

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 18 18 2 2 2 2 2 2 18 18 36 36 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 S3×D4 D4⋊2S3 kernel C62.256C23 C12⋊Dic3 C62⋊5C4 C2×C4×C3⋊S3 C2×C32⋊7D4 D4×C3×C6 C6×D4 C3×C12 C2×C3⋊S3 C2×C12 C22×C6 C3×C6 C12 C6 C6 # reps 1 1 2 1 2 1 4 2 2 4 8 2 16 4 4

Matrix representation of C62.256C23 in GL6(𝔽13)

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

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

C62.256C23 in GAP, Magma, Sage, TeX

`C_6^2._{256}C_2^3`
`% in TeX`

`G:=Group("C6^2.256C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,795);`
`// by ID`

`G=gap.SmallGroup(288,795);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,2693,9414]);`
`// 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,c*d=d*c,e*c*e=a^3*c,e*d*e=b^3*d>;`
`// generators/relations`

׿
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