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## G = C62⋊6Q8order 288 = 25·32

### 4th semidirect product of C62 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62⋊6Q8
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C62⋊6Q8
 Lower central C32 — C62 — C62⋊6Q8
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C626Q8
G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 732 in 222 conjugacy classes, 79 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×7], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C32, Dic3 [×20], C12 [×8], C2×C6 [×12], C2×C6 [×8], C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], Dic6 [×8], C2×Dic3 [×24], C2×C12 [×8], C22×C6 [×4], C22⋊Q8, C3⋊Dic3 [×2], C3⋊Dic3 [×3], C3×C12 [×2], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×4], C3×C22⋊C4 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12 [×2], C2×C62, Dic3.D4 [×4], C6.Dic6 [×2], C12⋊Dic3, C625C4, C32×C22⋊C4, C2×C324Q8, C22×C3⋊Dic3, C626Q8
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], Q8 [×2], C23, D6 [×12], C2×D4, C2×Q8, C4○D4, C3⋊S3, Dic6 [×8], C22×S3 [×4], C22⋊Q8, C2×C3⋊S3 [×3], C2×Dic6 [×4], S3×D4 [×4], D42S3 [×4], C324Q8 [×2], C22×C3⋊S3, Dic3.D4 [×4], C2×C324Q8, D4×C3⋊S3, C12.D6, C626Q8

Smallest permutation representation of C626Q8
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 7 38 20 24 4)(2 8 39 21 22 5)(3 9 37 19 23 6)(10 32 35 28 14 26)(11 33 36 29 15 27)(12 31 34 30 13 25)(16 61 59 57 66 68)(17 62 60 55 64 69)(18 63 58 56 65 67)(40 45 70 49 54 48)(41 43 71 50 52 46)(42 44 72 51 53 47)(73 80 143 76 83 140)(74 81 144 77 84 141)(75 82 139 78 79 142)(85 94 100 88 91 97)(86 95 101 89 92 98)(87 96 102 90 93 99)(103 118 111 106 115 114)(104 119 112 107 116 109)(105 120 113 108 117 110)(121 136 129 124 133 132)(122 137 130 125 134 127)(123 138 131 126 135 128)
(1 87 13 83)(2 85 14 81)(3 89 15 79)(4 99 30 76)(5 97 28 74)(6 101 29 78)(7 96 25 140)(8 94 26 144)(9 92 27 142)(10 77 39 100)(11 75 37 98)(12 73 38 102)(16 111 41 129)(17 109 42 127)(18 113 40 131)(19 86 33 82)(20 90 31 80)(21 88 32 84)(22 91 35 141)(23 95 36 139)(24 93 34 143)(43 124 61 106)(44 122 62 104)(45 126 63 108)(46 136 68 118)(47 134 69 116)(48 138 67 120)(49 128 56 110)(50 132 57 114)(51 130 55 112)(52 121 66 103)(53 125 64 107)(54 123 65 105)(58 117 70 135)(59 115 71 133)(60 119 72 137)
(1 46 13 68)(2 48 14 67)(3 47 15 69)(4 41 30 16)(5 40 28 18)(6 42 29 17)(7 52 25 66)(8 54 26 65)(9 53 27 64)(10 56 39 49)(11 55 37 51)(12 57 38 50)(19 72 33 60)(20 71 31 59)(21 70 32 58)(22 45 35 63)(23 44 36 62)(24 43 34 61)(73 132 102 114)(74 131 97 113)(75 130 98 112)(76 129 99 111)(77 128 100 110)(78 127 101 109)(79 134 89 116)(80 133 90 115)(81 138 85 120)(82 137 86 119)(83 136 87 118)(84 135 88 117)(91 108 141 126)(92 107 142 125)(93 106 143 124)(94 105 144 123)(95 104 139 122)(96 103 140 121)```

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

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

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

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 ··· 6T 12A ··· 12P 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 2 2 2 2 2 2 4 4 18 18 18 18 36 36 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 C4○D4 Dic6 S3×D4 D4⋊2S3 kernel C62⋊6Q8 C6.Dic6 C12⋊Dic3 C62⋊5C4 C32×C22⋊C4 C2×C32⋊4Q8 C22×C3⋊Dic3 C3×C22⋊C4 C3⋊Dic3 C62 C2×C12 C22×C6 C3×C6 C2×C6 C6 C6 # reps 1 2 1 1 1 1 1 4 2 2 8 4 2 16 4 4

Matrix representation of C626Q8 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 3 0 0 0 0 0 0 3 9
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 4 8

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

C626Q8 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_6Q_8`
`% in TeX`

`G:=Group("C6^2:6Q8");`
`// GroupNames label`

`G:=SmallGroup(288,735);`
`// by ID`

`G=gap.SmallGroup(288,735);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,254,219,58,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;`
`// generators/relations`

׿
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