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## G = C62.8Q8order 288 = 25·32

### 5th non-split extension by C62 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.8Q8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C32⋊4C8 — C62.8Q8
 Lower central C32 — C3×C6 — C3×C12 — C62.8Q8
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for C62.8Q8
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3b3c2, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a3b3c3 >

Subgroups: 220 in 90 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, M4(2), M4(2), C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C8.C4, C3×C12, C62, C2×C3⋊C8, C4.Dic3, C3×M4(2), C324C8, C324C8, C3×C24, C6×C12, C12.53D4, C2×C324C8, C12.58D6, C32×M4(2), C62.8Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, C3⋊S3, Dic6, C4×S3, C3⋊D4, C8.C4, C2×C3⋊S3, Dic3⋊C4, C324Q8, C4×C3⋊S3, C327D4, C12.53D4, C6.Dic6, C62.8Q8

Smallest permutation representation of C62.8Q8
On 144 points
Generators in S144
```(1 127 111)(2 124 112 6 128 108)(3 121 105)(4 126 106 8 122 110)(5 123 107)(7 125 109)(9 52 35)(10 49 36 14 53 40)(11 54 37)(12 51 38 16 55 34)(13 56 39)(15 50 33)(17 137 60 21 141 64)(18 142 61)(19 139 62 23 143 58)(20 144 63)(22 138 57)(24 140 59)(25 46 93 29 42 89)(26 43 94)(27 48 95 31 44 91)(28 45 96)(30 47 90)(32 41 92)(65 136 103)(66 133 104 70 129 100)(67 130 97)(68 135 98 72 131 102)(69 132 99)(71 134 101)(73 114 88)(74 119 81 78 115 85)(75 116 82)(76 113 83 80 117 87)(77 118 84)(79 120 86)
(1 130 86 5 134 82)(2 131 87 6 135 83)(3 132 88 7 136 84)(4 133 81 8 129 85)(9 43 24 13 47 20)(10 44 17 14 48 21)(11 45 18 15 41 22)(12 46 19 16 42 23)(25 58 38 29 62 34)(26 59 39 30 63 35)(27 60 40 31 64 36)(28 61 33 32 57 37)(49 91 137 53 95 141)(50 92 138 54 96 142)(51 93 139 55 89 143)(52 94 140 56 90 144)(65 118 105 69 114 109)(66 119 106 70 115 110)(67 120 107 71 116 111)(68 113 108 72 117 112)(73 125 103 77 121 99)(74 126 104 78 122 100)(75 127 97 79 123 101)(76 128 98 80 124 102)
(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 22 7 20 5 18 3 24)(2 17 4 19 6 21 8 23)(9 86 15 84 13 82 11 88)(10 81 12 83 14 85 16 87)(25 98 27 100 29 102 31 104)(26 97 32 103 30 101 28 99)(33 77 39 75 37 73 35 79)(34 80 36 74 38 76 40 78)(41 136 47 134 45 132 43 130)(42 131 44 133 46 135 48 129)(49 119 51 113 53 115 55 117)(50 118 56 116 54 114 52 120)(57 125 63 123 61 121 59 127)(58 128 60 122 62 124 64 126)(65 90 71 96 69 94 67 92)(66 93 68 95 70 89 72 91)(105 140 111 138 109 144 107 142)(106 143 108 137 110 139 112 141)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 3 3 3 3 4 4 4 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 2 2 2 2 2 1 1 2 2 2 2 2 4 4 4 4 4 4 18 18 18 18 36 36 2 ··· 2 4 4 4 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + - + - image C1 C2 C2 C2 C4 S3 D4 Q8 D6 C4×S3 C3⋊D4 Dic6 C8.C4 C12.53D4 kernel C62.8Q8 C2×C32⋊4C8 C12.58D6 C32×M4(2) C32⋊4C8 C3×M4(2) C3×C12 C62 C2×C12 C12 C12 C2×C6 C32 C3 # reps 1 1 1 1 4 4 1 1 4 8 8 8 4 8

Matrix representation of C62.8Q8 in GL6(𝔽73)

 0 72 0 0 0 0 1 72 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 52 72
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 68 3 0 0 0 0 7 5
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 10 32 0 0 0 0 22 63 0 0 0 0 0 0 63 0 0 0 0 0 20 51

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,52,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,68,7,0,0,0,0,3,5],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,10,22,0,0,0,0,32,63,0,0,0,0,0,0,63,20,0,0,0,0,0,51] >;`

C62.8Q8 in GAP, Magma, Sage, TeX

`C_6^2._8Q_8`
`% in TeX`

`G:=Group("C6^2.8Q8");`
`// GroupNames label`

`G:=SmallGroup(288,297);`
`// by ID`

`G=gap.SmallGroup(288,297);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,346,80,2693,9414]);`
`// Polycyclic`

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

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