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## G = C62.254C23order 288 = 25·32

### 99th non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.254C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.254C23
 Lower central C32 — C62 — C62.254C23
 Upper central C1 — C22 — C2×D4

Generators and relations for C62.254C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=b3, d2=a3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, ede=a3d >

Subgroups: 716 in 228 conjugacy classes, 77 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×4], C22, C22 [×6], C6 [×12], C6 [×8], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×16], C12 [×8], C2×C6 [×4], C2×C6 [×24], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×8], C2×Dic3 [×16], C2×C12 [×4], C3×D4 [×8], C22×C6 [×8], C4.4D4, C3⋊Dic3 [×4], C3×C12 [×2], C62, C62 [×6], C4×Dic3 [×4], C6.D4 [×16], C2×Dic6 [×4], C6×D4 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×4], C6×C12, D4×C32 [×2], C2×C62 [×2], C23.12D6 [×4], C4×C3⋊Dic3, C625C4 [×4], C2×C324Q8, D4×C3×C6, C62.254C23
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], D42S3 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C23.12D6 [×4], C12.D6 [×2], C2×C327D4, C62.254C23

Smallest permutation representation of C62.254C23
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 58 15 39 61)(2 36 59 16 40 62)(3 31 60 17 41 63)(4 32 55 18 42 64)(5 33 56 13 37 65)(6 34 57 14 38 66)(7 123 27 120 140 24)(8 124 28 115 141 19)(9 125 29 116 142 20)(10 126 30 117 143 21)(11 121 25 118 144 22)(12 122 26 119 139 23)(43 96 51 77 101 71)(44 91 52 78 102 72)(45 92 53 73 97 67)(46 93 54 74 98 68)(47 94 49 75 99 69)(48 95 50 76 100 70)(79 105 135 130 88 111)(80 106 136 131 89 112)(81 107 137 132 90 113)(82 108 138 127 85 114)(83 103 133 128 86 109)(84 104 134 129 87 110)
(1 90 15 107)(2 89 16 106)(3 88 17 105)(4 87 18 104)(5 86 13 103)(6 85 14 108)(7 76 120 48)(8 75 115 47)(9 74 116 46)(10 73 117 45)(11 78 118 44)(12 77 119 43)(19 99 28 94)(20 98 29 93)(21 97 30 92)(22 102 25 91)(23 101 26 96)(24 100 27 95)(31 130 41 79)(32 129 42 84)(33 128 37 83)(34 127 38 82)(35 132 39 81)(36 131 40 80)(49 141 69 124)(50 140 70 123)(51 139 71 122)(52 144 72 121)(53 143 67 126)(54 142 68 125)(55 134 64 110)(56 133 65 109)(57 138 66 114)(58 137 61 113)(59 136 62 112)(60 135 63 111)
(1 67 4 70)(2 68 5 71)(3 69 6 72)(7 132 10 129)(8 127 11 130)(9 128 12 131)(13 51 16 54)(14 52 17 49)(15 53 18 50)(19 138 22 135)(20 133 23 136)(21 134 24 137)(25 111 28 114)(26 112 29 109)(27 113 30 110)(31 47 34 44)(32 48 35 45)(33 43 36 46)(37 77 40 74)(38 78 41 75)(39 73 42 76)(55 95 58 92)(56 96 59 93)(57 91 60 94)(61 97 64 100)(62 98 65 101)(63 99 66 102)(79 115 82 118)(80 116 83 119)(81 117 84 120)(85 121 88 124)(86 122 89 125)(87 123 90 126)(103 139 106 142)(104 140 107 143)(105 141 108 144)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 81)(8 82)(9 83)(10 84)(11 79)(12 80)(13 51)(14 52)(15 53)(16 54)(17 49)(18 50)(19 114)(20 109)(21 110)(22 111)(23 112)(24 113)(25 135)(26 136)(27 137)(28 138)(29 133)(30 134)(31 47)(32 48)(33 43)(34 44)(35 45)(36 46)(37 77)(38 78)(39 73)(40 74)(41 75)(42 76)(55 95)(56 96)(57 91)(58 92)(59 93)(60 94)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(85 141)(86 142)(87 143)(88 144)(89 139)(90 140)(103 125)(104 126)(105 121)(106 122)(107 123)(108 124)(115 127)(116 128)(117 129)(118 130)(119 131)(120 132)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 6M ··· 6AB 12A ··· 12H order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 4 4 2 2 2 2 2 2 18 18 18 18 36 36 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 D6 D6 C4○D4 C3⋊D4 D4⋊2S3 kernel C62.254C23 C4×C3⋊Dic3 C62⋊5C4 C2×C32⋊4Q8 D4×C3×C6 C6×D4 C3×C12 C2×C12 C22×C6 C3×C6 C12 C6 # reps 1 1 4 1 1 4 2 4 8 4 16 8

Matrix representation of C62.254C23 in GL6(𝔽13)

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

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

C62.254C23 in GAP, Magma, Sage, TeX

`C_6^2._{254}C_2^3`
`% in TeX`

`G:=Group("C6^2.254C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,793);`
`// by ID`

`G=gap.SmallGroup(288,793);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,135,2693,9414]);`
`// Polycyclic`

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

׿
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