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## G = C62⋊10Q8order 288 = 25·32

### 3rd semidirect product of C62 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62⋊10Q8
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C2×C32⋊4Q8 — C62⋊10Q8
 Lower central C32 — C62 — C62⋊10Q8
 Upper central C1 — C22 — C22×C4

Generators and relations for C6210Q8
G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1b3, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 668 in 222 conjugacy classes, 89 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C32, Dic3 [×16], C12 [×8], C12 [×4], C2×C6 [×12], C2×C6 [×8], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], Dic6 [×8], C2×Dic3 [×16], C2×C12 [×8], C2×C12 [×8], C22×C6 [×4], C22⋊Q8, C3⋊Dic3 [×4], C3×C12 [×2], C3×C12, C62, C62 [×2], C62 [×2], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×8], C2×Dic6 [×4], C22×C12 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×4], C6×C12 [×2], C6×C12 [×2], C2×C62, C12.48D4 [×4], C6.Dic6 [×2], C12⋊Dic3, C625C4 [×2], C2×C324Q8, C2×C6×C12, C6210Q8
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], C3⋊D4 [×8], C22×S3 [×4], C22⋊Q8, C2×C3⋊S3 [×3], C2×Dic6 [×4], C4○D12 [×4], C2×C3⋊D4 [×4], C324Q8 [×2], C327D4 [×2], C22×C3⋊S3, C12.48D4 [×4], C2×C324Q8, C12.59D6, C2×C327D4, C6210Q8

Smallest permutation representation of C6210Q8
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 25 6 32 14 19)(2 26 4 33 15 20)(3 27 5 31 13 21)(7 18 30 22 36 12)(8 16 28 23 34 10)(9 17 29 24 35 11)(37 49 57 65 60 71)(38 50 55 66 58 72)(39 51 56 64 59 70)(40 47 68 53 63 43)(41 48 69 54 61 44)(42 46 67 52 62 45)(73 114 96 76 111 93)(74 109 91 77 112 94)(75 110 92 78 113 95)(79 143 123 82 140 126)(80 144 124 83 141 121)(81 139 125 84 142 122)(85 138 98 88 135 101)(86 133 99 89 136 102)(87 134 100 90 137 97)(103 129 116 106 132 119)(104 130 117 107 127 120)(105 131 118 108 128 115)
(1 68 23 59)(2 69 24 60)(3 67 22 58)(4 61 11 37)(5 62 12 38)(6 63 10 39)(7 50 31 45)(8 51 32 43)(9 49 33 44)(13 42 18 55)(14 40 16 56)(15 41 17 57)(19 47 28 64)(20 48 29 65)(21 46 30 66)(25 53 34 70)(26 54 35 71)(27 52 36 72)(73 121 128 97)(74 122 129 98)(75 123 130 99)(76 124 131 100)(77 125 132 101)(78 126 127 102)(79 120 86 113)(80 115 87 114)(81 116 88 109)(82 117 89 110)(83 118 90 111)(84 119 85 112)(91 139 106 135)(92 140 107 136)(93 141 108 137)(94 142 103 138)(95 143 104 133)(96 144 105 134)
(1 129 23 74)(2 131 24 76)(3 127 22 78)(4 115 11 114)(5 117 12 110)(6 119 10 112)(7 75 31 130)(8 77 32 132)(9 73 33 128)(13 104 18 95)(14 106 16 91)(15 108 17 93)(19 116 28 109)(20 118 29 111)(21 120 30 113)(25 103 34 94)(26 105 35 96)(27 107 36 92)(37 87 61 80)(38 89 62 82)(39 85 63 84)(40 139 56 135)(41 141 57 137)(42 143 55 133)(43 125 51 101)(44 121 49 97)(45 123 50 99)(46 79 66 86)(47 81 64 88)(48 83 65 90)(52 140 72 136)(53 142 70 138)(54 144 71 134)(58 102 67 126)(59 98 68 122)(60 100 69 124)```

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6AB 12A ··· 12AF 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 2 2 2 2 2 2 2 2 2 2 36 36 36 36 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D12 kernel C62⋊10Q8 C6.Dic6 C12⋊Dic3 C62⋊5C4 C2×C32⋊4Q8 C2×C6×C12 C22×C12 C3×C12 C62 C2×C12 C22×C6 C3×C6 C12 C2×C6 C6 # reps 1 2 1 2 1 1 4 2 2 8 4 2 16 16 16

Matrix representation of C6210Q8 in GL6(𝔽13)

 1 0 0 0 0 0 12 12 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 9
,
 5 0 0 0 0 0 8 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 12 11 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0

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

C6210Q8 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_{10}Q_8`
`% in TeX`

`G:=Group("C6^2:10Q8");`
`// GroupNames label`

`G:=SmallGroup(288,781);`
`// by ID`

`G=gap.SmallGroup(288,781);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,254,2693,9414]);`
`// Polycyclic`

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

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