metabelian, supersoluble, monomial
Aliases: C62⋊7C8, (C6×C12).20C4, (C2×C12).421D6, (C3×C12).172D4, (C2×C62).11C4, C32⋊10(C22⋊C8), C62.103(C2×C4), (C22×C12).35S3, (C2×C12).13Dic3, (C3×C6).23M4(2), C3⋊2(C12.55D4), C12.133(C3⋊D4), (C6×C12).343C22, C2.1(C62⋊5C4), C22⋊2(C32⋊4C8), C4.30(C32⋊7D4), C6.10(C4.Dic3), C23.3(C3⋊Dic3), (C22×C6).15Dic3, C6.21(C6.D4), C2.3(C12.58D6), (C2×C6)⋊4(C3⋊C8), C6.16(C2×C3⋊C8), (C2×C6×C12).19C2, (C3×C6).46(C2×C8), C2.5(C2×C32⋊4C8), (C2×C32⋊4C8)⋊16C2, (C22×C4).3(C3⋊S3), (C2×C4).3(C3⋊Dic3), (C2×C6).45(C2×Dic3), C22.9(C2×C3⋊Dic3), (C3×C6).70(C22⋊C4), (C2×C4).94(C2×C3⋊S3), SmallGroup(288,305)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊7C8
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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C32, C12, C12, C2×C6, C2×C6, C2×C8, C22×C4, C3×C6, C3×C6, C3⋊C8, C2×C12, C2×C12, C22×C6, C22⋊C8, C3×C12, C3×C12, C62, C62, C62, C2×C3⋊C8, C22×C12, C32⋊4C8, C6×C12, C6×C12, C2×C62, C12.55D4, C2×C32⋊4C8, C2×C6×C12, C62⋊7C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), C3⋊S3, C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C3⋊Dic3, C2×C3⋊S3, C2×C3⋊C8, C4.Dic3, C6.D4, C32⋊4C8, C2×C3⋊Dic3, C32⋊7D4, C12.55D4, C2×C32⋊4C8, C12.58D6, C62⋊5C4, C62⋊7C8
►On 144 points
Generators in S144
(1 40 141)(2 31 33 82 142 57)(3 34 143)(4 25 35 84 144 59)(5 36 137)(6 27 37 86 138 61)(7 38 139)(8 29 39 88 140 63)(9 122 94)(10 118 123 51 95 75)(11 124 96)(12 120 125 53 89 77)(13 126 90)(14 114 127 55 91 79)(15 128 92)(16 116 121 49 93 73)(17 46 70 111 129 101)(18 71 130)(19 48 72 105 131 103)(20 65 132)(21 42 66 107 133 97)(22 67 134)(23 44 68 109 135 99)(24 69 136)(26 85 60)(28 87 62)(30 81 64)(32 83 58)(41 106 104)(43 108 98)(45 110 100)(47 112 102)(50 74 117)(52 76 119)(54 78 113)(56 80 115)
(1 41 9 81 132 50)(2 51 133 82 10 42)(3 43 11 83 134 52)(4 53 135 84 12 44)(5 45 13 85 136 54)(6 55 129 86 14 46)(7 47 15 87 130 56)(8 49 131 88 16 48)(17 61 127 111 37 79)(18 80 38 112 128 62)(19 63 121 105 39 73)(20 74 40 106 122 64)(21 57 123 107 33 75)(22 76 34 108 124 58)(23 59 125 109 35 77)(24 78 36 110 126 60)(25 89 99 144 120 68)(26 69 113 137 100 90)(27 91 101 138 114 70)(28 71 115 139 102 92)(29 93 103 140 116 72)(30 65 117 141 104 94)(31 95 97 142 118 66)(32 67 119 143 98 96)
(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)| (1,40,141)(2,31,33,82,142,57)(3,34,143)(4,25,35,84,144,59)(5,36,137)(6,27,37,86,138,61)(7,38,139)(8,29,39,88,140,63)(9,122,94)(10,118,123,51,95,75)(11,124,96)(12,120,125,53,89,77)(13,126,90)(14,114,127,55,91,79)(15,128,92)(16,116,121,49,93,73)(17,46,70,111,129,101)(18,71,130)(19,48,72,105,131,103)(20,65,132)(21,42,66,107,133,97)(22,67,134)(23,44,68,109,135,99)(24,69,136)(26,85,60)(28,87,62)(30,81,64)(32,83,58)(41,106,104)(43,108,98)(45,110,100)(47,112,102)(50,74,117)(52,76,119)(54,78,113)(56,80,115), (1,41,9,81,132,50)(2,51,133,82,10,42)(3,43,11,83,134,52)(4,53,135,84,12,44)(5,45,13,85,136,54)(6,55,129,86,14,46)(7,47,15,87,130,56)(8,49,131,88,16,48)(17,61,127,111,37,79)(18,80,38,112,128,62)(19,63,121,105,39,73)(20,74,40,106,122,64)(21,57,123,107,33,75)(22,76,34,108,124,58)(23,59,125,109,35,77)(24,78,36,110,126,60)(25,89,99,144,120,68)(26,69,113,137,100,90)(27,91,101,138,114,70)(28,71,115,139,102,92)(29,93,103,140,116,72)(30,65,117,141,104,94)(31,95,97,142,118,66)(32,67,119,143,98,96), (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( (1,40,141)(2,31,33,82,142,57)(3,34,143)(4,25,35,84,144,59)(5,36,137)(6,27,37,86,138,61)(7,38,139)(8,29,39,88,140,63)(9,122,94)(10,118,123,51,95,75)(11,124,96)(12,120,125,53,89,77)(13,126,90)(14,114,127,55,91,79)(15,128,92)(16,116,121,49,93,73)(17,46,70,111,129,101)(18,71,130)(19,48,72,105,131,103)(20,65,132)(21,42,66,107,133,97)(22,67,134)(23,44,68,109,135,99)(24,69,136)(26,85,60)(28,87,62)(30,81,64)(32,83,58)(41,106,104)(43,108,98)(45,110,100)(47,112,102)(50,74,117)(52,76,119)(54,78,113)(56,80,115), (1,41,9,81,132,50)(2,51,133,82,10,42)(3,43,11,83,134,52)(4,53,135,84,12,44)(5,45,13,85,136,54)(6,55,129,86,14,46)(7,47,15,87,130,56)(8,49,131,88,16,48)(17,61,127,111,37,79)(18,80,38,112,128,62)(19,63,121,105,39,73)(20,74,40,106,122,64)(21,57,123,107,33,75)(22,76,34,108,124,58)(23,59,125,109,35,77)(24,78,36,110,126,60)(25,89,99,144,120,68)(26,69,113,137,100,90)(27,91,101,138,114,70)(28,71,115,139,102,92)(29,93,103,140,116,72)(30,65,117,141,104,94)(31,95,97,142,118,66)(32,67,119,143,98,96), (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([[(1,40,141),(2,31,33,82,142,57),(3,34,143),(4,25,35,84,144,59),(5,36,137),(6,27,37,86,138,61),(7,38,139),(8,29,39,88,140,63),(9,122,94),(10,118,123,51,95,75),(11,124,96),(12,120,125,53,89,77),(13,126,90),(14,114,127,55,91,79),(15,128,92),(16,116,121,49,93,73),(17,46,70,111,129,101),(18,71,130),(19,48,72,105,131,103),(20,65,132),(21,42,66,107,133,97),(22,67,134),(23,44,68,109,135,99),(24,69,136),(26,85,60),(28,87,62),(30,81,64),(32,83,58),(41,106,104),(43,108,98),(45,110,100),(47,112,102),(50,74,117),(52,76,119),(54,78,113),(56,80,115)], [(1,41,9,81,132,50),(2,51,133,82,10,42),(3,43,11,83,134,52),(4,53,135,84,12,44),(5,45,13,85,136,54),(6,55,129,86,14,46),(7,47,15,87,130,56),(8,49,131,88,16,48),(17,61,127,111,37,79),(18,80,38,112,128,62),(19,63,121,105,39,73),(20,74,40,106,122,64),(21,57,123,107,33,75),(22,76,34,108,124,58),(23,59,125,109,35,77),(24,78,36,110,126,60),(25,89,99,144,120,68),(26,69,113,137,100,90),(27,91,101,138,114,70),(28,71,115,139,102,92),(29,93,103,140,116,72),(30,65,117,141,104,94),(31,95,97,142,118,66),(32,67,119,143,98,96)], [(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 C62⋊7C8 ►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] >;
C62⋊7C8 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