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## G = C62⋊7C8order 288 = 25·32

### 3rd semidirect product of C62 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62⋊7C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C32⋊4C8 — C62⋊7C8
 Lower central C32 — C3×C6 — C62⋊7C8
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C627C8
G = < a,b,c | a6=b6=c8=1, ab=ba, cac-1=a-1b3, cbc-1=b-1 >

Subgroups: 316 in 150 conjugacy classes, 81 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4, C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C32, C12 [×8], C12 [×4], C2×C6 [×12], C2×C6 [×8], C2×C8 [×2], C22×C4, C3×C6 [×3], C3×C6 [×2], C3⋊C8 [×8], C2×C12 [×8], C2×C12 [×8], C22×C6 [×4], C22⋊C8, C3×C12 [×2], C3×C12, C62, C62 [×2], C62 [×2], C2×C3⋊C8 [×8], C22×C12 [×4], C324C8 [×2], C6×C12 [×2], C6×C12 [×2], C2×C62, C12.55D4 [×4], C2×C324C8 [×2], C2×C6×C12, C627C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, D4 [×2], Dic3 [×8], D6 [×4], C22⋊C4, C2×C8, M4(2), C3⋊S3, C3⋊C8 [×8], C2×Dic3 [×4], C3⋊D4 [×8], C22⋊C8, C3⋊Dic3 [×2], C2×C3⋊S3, C2×C3⋊C8 [×4], C4.Dic3 [×4], C6.D4 [×4], C324C8 [×2], C2×C3⋊Dic3, C327D4 [×2], C12.55D4 [×4], C2×C324C8, C12.58D6, C625C4, C627C8

Smallest permutation representation of C627C8
On 144 points
Generators in S144
```(1 112 141)(2 51 105 46 142 125)(3 106 143)(4 53 107 48 144 127)(5 108 137)(6 55 109 42 138 121)(7 110 139)(8 49 111 44 140 123)(9 96 97)(10 118 89 17 98 75)(11 90 99)(12 120 91 19 100 77)(13 92 101)(14 114 93 21 102 79)(15 94 103)(16 116 95 23 104 73)(18 76 119)(20 78 113)(22 80 115)(24 74 117)(25 135 88 68 37 59)(26 81 38)(27 129 82 70 39 61)(28 83 40)(29 131 84 72 33 63)(30 85 34)(31 133 86 66 35 57)(32 87 36)(41 128 54)(43 122 56)(45 124 50)(47 126 52)(58 134 67)(60 136 69)(62 130 71)(64 132 65)
(1 85 9 45 64 24)(2 17 57 46 10 86)(3 87 11 47 58 18)(4 19 59 48 12 88)(5 81 13 41 60 20)(6 21 61 42 14 82)(7 83 15 43 62 22)(8 23 63 44 16 84)(25 144 120 68 53 100)(26 101 54 69 113 137)(27 138 114 70 55 102)(28 103 56 71 115 139)(29 140 116 72 49 104)(30 97 50 65 117 141)(31 142 118 66 51 98)(32 99 52 67 119 143)(33 111 73 131 123 95)(34 96 124 132 74 112)(35 105 75 133 125 89)(36 90 126 134 76 106)(37 107 77 135 127 91)(38 92 128 136 78 108)(39 109 79 129 121 93)(40 94 122 130 80 110)
(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)```

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6AB 8A ··· 8H 12A ··· 12AF order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 ··· 2 18 ··· 18 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C4 C4 C8 S3 D4 Dic3 D6 Dic3 M4(2) C3⋊D4 C3⋊C8 C4.Dic3 kernel C62⋊7C8 C2×C32⋊4C8 C2×C6×C12 C6×C12 C2×C62 C62 C22×C12 C3×C12 C2×C12 C2×C12 C22×C6 C3×C6 C12 C2×C6 C6 # reps 1 2 1 2 2 8 4 2 4 4 4 2 16 16 16

Matrix representation of C627C8 in GL4(𝔽73) generated by

 8 0 0 0 0 64 0 0 0 0 8 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 65
,
 0 1 0 0 27 0 0 0 0 0 0 1 0 0 27 0
`G:=sub<GL(4,GF(73))| [8,0,0,0,0,64,0,0,0,0,8,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,65],[0,27,0,0,1,0,0,0,0,0,0,27,0,0,1,0] >;`

C627C8 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_7C_8`
`% in TeX`

`G:=Group("C6^2:7C8");`
`// GroupNames label`

`G:=SmallGroup(288,305);`
`// by ID`

`G=gap.SmallGroup(288,305);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,100,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^6=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b^3,c*b*c^-1=b^-1>;`
`// generators/relations`

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