metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C21⋊Q8, C7⋊1Dic6, C6.7D14, C14.7D6, C3⋊1Dic14, Dic7.S3, Dic3.D7, C42.7C22, Dic21.2C2, C2.7(S3×D7), (C3×Dic7).1C2, (C7×Dic3).1C2, SmallGroup(168,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C21⋊Q8
G = < a,b,c | a21=b4=1, c2=b2, bab-1=a8, cac-1=a13, cbc-1=b-1 >
Character table of C21⋊Q8
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 7A | 7B | 7C | 12A | 12B | 14A | 14B | 14C | 21A | 21B | 21C | 28A | 28B | 28C | 28D | 28E | 28F | 42A | 42B | 42C | |
| size | 1 | 1 | 2 | 6 | 14 | 42 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | -1 | 0 | -2 | 0 | -1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ7 | 2 | 2 | -1 | 0 | 2 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ12 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ13 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | 2 | 2 | 2 | -√3 | √3 | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | 2 | 2 | 2 | √3 | -√3 | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ4ζ76+ζ4ζ7 | -ζ43ζ75+ζ43ζ72 | ζ43ζ75-ζ43ζ72 | ζ4ζ76-ζ4ζ7 | -ζ4ζ74+ζ4ζ73 | ζ4ζ74-ζ4ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | symplectic lifted from Dic14, Schur index 2 |
| ρ17 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ4ζ74-ζ4ζ73 | ζ4ζ76-ζ4ζ7 | -ζ4ζ76+ζ4ζ7 | -ζ4ζ74+ζ4ζ73 | -ζ43ζ75+ζ43ζ72 | ζ43ζ75-ζ43ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | symplectic lifted from Dic14, Schur index 2 |
| ρ18 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ4ζ74+ζ4ζ73 | -ζ4ζ76+ζ4ζ7 | ζ4ζ76-ζ4ζ7 | ζ4ζ74-ζ4ζ73 | ζ43ζ75-ζ43ζ72 | -ζ43ζ75+ζ43ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | symplectic lifted from Dic14, Schur index 2 |
| ρ19 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ4ζ76-ζ4ζ7 | ζ43ζ75-ζ43ζ72 | -ζ43ζ75+ζ43ζ72 | -ζ4ζ76+ζ4ζ7 | ζ4ζ74-ζ4ζ73 | -ζ4ζ74+ζ4ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | symplectic lifted from Dic14, Schur index 2 |
| ρ20 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ43ζ75-ζ43ζ72 | -ζ4ζ74+ζ4ζ73 | ζ4ζ74-ζ4ζ73 | -ζ43ζ75+ζ43ζ72 | ζ4ζ76-ζ4ζ7 | -ζ4ζ76+ζ4ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | symplectic lifted from Dic14, Schur index 2 |
| ρ21 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ43ζ75+ζ43ζ72 | ζ4ζ74-ζ4ζ73 | -ζ4ζ74+ζ4ζ73 | ζ43ζ75-ζ43ζ72 | -ζ4ζ76+ζ4ζ7 | ζ4ζ76-ζ4ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | symplectic lifted from Dic14, Schur index 2 |
| ρ22 | 4 | 4 | -2 | 0 | 0 | 0 | -2 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ23 | 4 | 4 | -2 | 0 | 0 | 0 | -2 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ24 | 4 | 4 | -2 | 0 | 0 | 0 | -2 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
| ρ25 | 4 | -4 | -2 | 0 | 0 | 0 | 2 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | -2 | 0 | 0 | 0 | 2 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | -2 | 0 | 0 | 0 | 2 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | symplectic faithful, Schur index 2 |
►Regular action on 168 points
Generators in S168
(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)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 80 40 63)(2 67 41 50)(3 75 42 58)(4 83 22 45)(5 70 23 53)(6 78 24 61)(7 65 25 48)(8 73 26 56)(9 81 27 43)(10 68 28 51)(11 76 29 59)(12 84 30 46)(13 71 31 54)(14 79 32 62)(15 66 33 49)(16 74 34 57)(17 82 35 44)(18 69 36 52)(19 77 37 60)(20 64 38 47)(21 72 39 55)(85 140 123 161)(86 127 124 148)(87 135 125 156)(88 143 126 164)(89 130 106 151)(90 138 107 159)(91 146 108 167)(92 133 109 154)(93 141 110 162)(94 128 111 149)(95 136 112 157)(96 144 113 165)(97 131 114 152)(98 139 115 160)(99 147 116 168)(100 134 117 155)(101 142 118 163)(102 129 119 150)(103 137 120 158)(104 145 121 166)(105 132 122 153)
(1 112 40 95)(2 125 41 87)(3 117 42 100)(4 109 22 92)(5 122 23 105)(6 114 24 97)(7 106 25 89)(8 119 26 102)(9 111 27 94)(10 124 28 86)(11 116 29 99)(12 108 30 91)(13 121 31 104)(14 113 32 96)(15 126 33 88)(16 118 34 101)(17 110 35 93)(18 123 36 85)(19 115 37 98)(20 107 38 90)(21 120 39 103)(43 149 81 128)(44 162 82 141)(45 154 83 133)(46 167 84 146)(47 159 64 138)(48 151 65 130)(49 164 66 143)(50 156 67 135)(51 148 68 127)(52 161 69 140)(53 153 70 132)(54 166 71 145)(55 158 72 137)(56 150 73 129)(57 163 74 142)(58 155 75 134)(59 168 76 147)(60 160 77 139)(61 152 78 131)(62 165 79 144)(63 157 80 136)
G:=sub<Sym(168)| (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,80,40,63)(2,67,41,50)(3,75,42,58)(4,83,22,45)(5,70,23,53)(6,78,24,61)(7,65,25,48)(8,73,26,56)(9,81,27,43)(10,68,28,51)(11,76,29,59)(12,84,30,46)(13,71,31,54)(14,79,32,62)(15,66,33,49)(16,74,34,57)(17,82,35,44)(18,69,36,52)(19,77,37,60)(20,64,38,47)(21,72,39,55)(85,140,123,161)(86,127,124,148)(87,135,125,156)(88,143,126,164)(89,130,106,151)(90,138,107,159)(91,146,108,167)(92,133,109,154)(93,141,110,162)(94,128,111,149)(95,136,112,157)(96,144,113,165)(97,131,114,152)(98,139,115,160)(99,147,116,168)(100,134,117,155)(101,142,118,163)(102,129,119,150)(103,137,120,158)(104,145,121,166)(105,132,122,153), (1,112,40,95)(2,125,41,87)(3,117,42,100)(4,109,22,92)(5,122,23,105)(6,114,24,97)(7,106,25,89)(8,119,26,102)(9,111,27,94)(10,124,28,86)(11,116,29,99)(12,108,30,91)(13,121,31,104)(14,113,32,96)(15,126,33,88)(16,118,34,101)(17,110,35,93)(18,123,36,85)(19,115,37,98)(20,107,38,90)(21,120,39,103)(43,149,81,128)(44,162,82,141)(45,154,83,133)(46,167,84,146)(47,159,64,138)(48,151,65,130)(49,164,66,143)(50,156,67,135)(51,148,68,127)(52,161,69,140)(53,153,70,132)(54,166,71,145)(55,158,72,137)(56,150,73,129)(57,163,74,142)(58,155,75,134)(59,168,76,147)(60,160,77,139)(61,152,78,131)(62,165,79,144)(63,157,80,136)>;
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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,80,40,63)(2,67,41,50)(3,75,42,58)(4,83,22,45)(5,70,23,53)(6,78,24,61)(7,65,25,48)(8,73,26,56)(9,81,27,43)(10,68,28,51)(11,76,29,59)(12,84,30,46)(13,71,31,54)(14,79,32,62)(15,66,33,49)(16,74,34,57)(17,82,35,44)(18,69,36,52)(19,77,37,60)(20,64,38,47)(21,72,39,55)(85,140,123,161)(86,127,124,148)(87,135,125,156)(88,143,126,164)(89,130,106,151)(90,138,107,159)(91,146,108,167)(92,133,109,154)(93,141,110,162)(94,128,111,149)(95,136,112,157)(96,144,113,165)(97,131,114,152)(98,139,115,160)(99,147,116,168)(100,134,117,155)(101,142,118,163)(102,129,119,150)(103,137,120,158)(104,145,121,166)(105,132,122,153), (1,112,40,95)(2,125,41,87)(3,117,42,100)(4,109,22,92)(5,122,23,105)(6,114,24,97)(7,106,25,89)(8,119,26,102)(9,111,27,94)(10,124,28,86)(11,116,29,99)(12,108,30,91)(13,121,31,104)(14,113,32,96)(15,126,33,88)(16,118,34,101)(17,110,35,93)(18,123,36,85)(19,115,37,98)(20,107,38,90)(21,120,39,103)(43,149,81,128)(44,162,82,141)(45,154,83,133)(46,167,84,146)(47,159,64,138)(48,151,65,130)(49,164,66,143)(50,156,67,135)(51,148,68,127)(52,161,69,140)(53,153,70,132)(54,166,71,145)(55,158,72,137)(56,150,73,129)(57,163,74,142)(58,155,75,134)(59,168,76,147)(60,160,77,139)(61,152,78,131)(62,165,79,144)(63,157,80,136) );
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),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,80,40,63),(2,67,41,50),(3,75,42,58),(4,83,22,45),(5,70,23,53),(6,78,24,61),(7,65,25,48),(8,73,26,56),(9,81,27,43),(10,68,28,51),(11,76,29,59),(12,84,30,46),(13,71,31,54),(14,79,32,62),(15,66,33,49),(16,74,34,57),(17,82,35,44),(18,69,36,52),(19,77,37,60),(20,64,38,47),(21,72,39,55),(85,140,123,161),(86,127,124,148),(87,135,125,156),(88,143,126,164),(89,130,106,151),(90,138,107,159),(91,146,108,167),(92,133,109,154),(93,141,110,162),(94,128,111,149),(95,136,112,157),(96,144,113,165),(97,131,114,152),(98,139,115,160),(99,147,116,168),(100,134,117,155),(101,142,118,163),(102,129,119,150),(103,137,120,158),(104,145,121,166),(105,132,122,153)], [(1,112,40,95),(2,125,41,87),(3,117,42,100),(4,109,22,92),(5,122,23,105),(6,114,24,97),(7,106,25,89),(8,119,26,102),(9,111,27,94),(10,124,28,86),(11,116,29,99),(12,108,30,91),(13,121,31,104),(14,113,32,96),(15,126,33,88),(16,118,34,101),(17,110,35,93),(18,123,36,85),(19,115,37,98),(20,107,38,90),(21,120,39,103),(43,149,81,128),(44,162,82,141),(45,154,83,133),(46,167,84,146),(47,159,64,138),(48,151,65,130),(49,164,66,143),(50,156,67,135),(51,148,68,127),(52,161,69,140),(53,153,70,132),(54,166,71,145),(55,158,72,137),(56,150,73,129),(57,163,74,142),(58,155,75,134),(59,168,76,147),(60,160,77,139),(61,152,78,131),(62,165,79,144),(63,157,80,136)]])
C21⋊Q8 is a maximal subgroup of
D7×Dic6 S3×Dic14 D21⋊Q8 D6.D14 Dic7.D6 C42.C23 Dic3.D14
C21⋊Q8 is a maximal quotient of C42.Q8 Dic21⋊C4 C14.Dic6
Matrix representation of C21⋊Q8 ►in GL4(𝔽337) generated by
| 227 | 110 | 0 | 0 |
| 117 | 76 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 336 | 336 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 142 | 218 |
| 0 | 0 | 76 | 195 |
| 83 | 115 | 0 | 0 |
| 113 | 254 | 0 | 0 |
| 0 | 0 | 322 | 307 |
| 0 | 0 | 30 | 15 |
G:=sub<GL(4,GF(337))| [227,117,0,0,110,76,0,0,0,0,0,336,0,0,1,336],[1,0,0,0,0,1,0,0,0,0,142,76,0,0,218,195],[83,113,0,0,115,254,0,0,0,0,322,30,0,0,307,15] >;
C21⋊Q8 in GAP, Magma, Sage, TeX
C_{21}\rtimes Q_8 % in TeX
G:=Group("C21:Q8"); // GroupNames label
G:=SmallGroup(168,18);
// by ID
G=gap.SmallGroup(168,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,20,61,26,168,3604]);
// Polycyclic
G:=Group<a,b,c|a^21=b^4=1,c^2=b^2,b*a*b^-1=a^8,c*a*c^-1=a^13,c*b*c^-1=b^-1>;
// generators/relations
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Subgroup lattice of C21⋊Q8 in TeX
Character table of C21⋊Q8 in TeX