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## G = C21⋊Q8order 168 = 23·3·7

### The semidirect product of C21 and Q8 acting via Q8/C2=C22

Aliases: C21⋊Q8, C71Dic6, C6.7D14, C14.7D6, C31Dic14, 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

 Derived series C1 — C42 — C21⋊Q8
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — C21⋊Q8
 Lower central C21 — C42 — C21⋊Q8
 Upper central C1 — C2

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

Smallest permutation representation of C21⋊Q8
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 76 23 46)(2 84 24 54)(3 71 25 62)(4 79 26 49)(5 66 27 57)(6 74 28 44)(7 82 29 52)(8 69 30 60)(9 77 31 47)(10 64 32 55)(11 72 33 63)(12 80 34 50)(13 67 35 58)(14 75 36 45)(15 83 37 53)(16 70 38 61)(17 78 39 48)(18 65 40 56)(19 73 41 43)(20 81 42 51)(21 68 22 59)(85 146 113 165)(86 133 114 152)(87 141 115 160)(88 128 116 168)(89 136 117 155)(90 144 118 163)(91 131 119 150)(92 139 120 158)(93 147 121 166)(94 134 122 153)(95 142 123 161)(96 129 124 148)(97 137 125 156)(98 145 126 164)(99 132 106 151)(100 140 107 159)(101 127 108 167)(102 135 109 154)(103 143 110 162)(104 130 111 149)(105 138 112 157)
(1 113 23 85)(2 126 24 98)(3 118 25 90)(4 110 26 103)(5 123 27 95)(6 115 28 87)(7 107 29 100)(8 120 30 92)(9 112 31 105)(10 125 32 97)(11 117 33 89)(12 109 34 102)(13 122 35 94)(14 114 36 86)(15 106 37 99)(16 119 38 91)(17 111 39 104)(18 124 40 96)(19 116 41 88)(20 108 42 101)(21 121 22 93)(43 168 73 128)(44 160 74 141)(45 152 75 133)(46 165 76 146)(47 157 77 138)(48 149 78 130)(49 162 79 143)(50 154 80 135)(51 167 81 127)(52 159 82 140)(53 151 83 132)(54 164 84 145)(55 156 64 137)(56 148 65 129)(57 161 66 142)(58 153 67 134)(59 166 68 147)(60 158 69 139)(61 150 70 131)(62 163 71 144)(63 155 72 136)```

`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), (1,76,23,46)(2,84,24,54)(3,71,25,62)(4,79,26,49)(5,66,27,57)(6,74,28,44)(7,82,29,52)(8,69,30,60)(9,77,31,47)(10,64,32,55)(11,72,33,63)(12,80,34,50)(13,67,35,58)(14,75,36,45)(15,83,37,53)(16,70,38,61)(17,78,39,48)(18,65,40,56)(19,73,41,43)(20,81,42,51)(21,68,22,59)(85,146,113,165)(86,133,114,152)(87,141,115,160)(88,128,116,168)(89,136,117,155)(90,144,118,163)(91,131,119,150)(92,139,120,158)(93,147,121,166)(94,134,122,153)(95,142,123,161)(96,129,124,148)(97,137,125,156)(98,145,126,164)(99,132,106,151)(100,140,107,159)(101,127,108,167)(102,135,109,154)(103,143,110,162)(104,130,111,149)(105,138,112,157), (1,113,23,85)(2,126,24,98)(3,118,25,90)(4,110,26,103)(5,123,27,95)(6,115,28,87)(7,107,29,100)(8,120,30,92)(9,112,31,105)(10,125,32,97)(11,117,33,89)(12,109,34,102)(13,122,35,94)(14,114,36,86)(15,106,37,99)(16,119,38,91)(17,111,39,104)(18,124,40,96)(19,116,41,88)(20,108,42,101)(21,121,22,93)(43,168,73,128)(44,160,74,141)(45,152,75,133)(46,165,76,146)(47,157,77,138)(48,149,78,130)(49,162,79,143)(50,154,80,135)(51,167,81,127)(52,159,82,140)(53,151,83,132)(54,164,84,145)(55,156,64,137)(56,148,65,129)(57,161,66,142)(58,153,67,134)(59,166,68,147)(60,158,69,139)(61,150,70,131)(62,163,71,144)(63,155,72,136)>;`

`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), (1,76,23,46)(2,84,24,54)(3,71,25,62)(4,79,26,49)(5,66,27,57)(6,74,28,44)(7,82,29,52)(8,69,30,60)(9,77,31,47)(10,64,32,55)(11,72,33,63)(12,80,34,50)(13,67,35,58)(14,75,36,45)(15,83,37,53)(16,70,38,61)(17,78,39,48)(18,65,40,56)(19,73,41,43)(20,81,42,51)(21,68,22,59)(85,146,113,165)(86,133,114,152)(87,141,115,160)(88,128,116,168)(89,136,117,155)(90,144,118,163)(91,131,119,150)(92,139,120,158)(93,147,121,166)(94,134,122,153)(95,142,123,161)(96,129,124,148)(97,137,125,156)(98,145,126,164)(99,132,106,151)(100,140,107,159)(101,127,108,167)(102,135,109,154)(103,143,110,162)(104,130,111,149)(105,138,112,157), (1,113,23,85)(2,126,24,98)(3,118,25,90)(4,110,26,103)(5,123,27,95)(6,115,28,87)(7,107,29,100)(8,120,30,92)(9,112,31,105)(10,125,32,97)(11,117,33,89)(12,109,34,102)(13,122,35,94)(14,114,36,86)(15,106,37,99)(16,119,38,91)(17,111,39,104)(18,124,40,96)(19,116,41,88)(20,108,42,101)(21,121,22,93)(43,168,73,128)(44,160,74,141)(45,152,75,133)(46,165,76,146)(47,157,77,138)(48,149,78,130)(49,162,79,143)(50,154,80,135)(51,167,81,127)(52,159,82,140)(53,151,83,132)(54,164,84,145)(55,156,64,137)(56,148,65,129)(57,161,66,142)(58,153,67,134)(59,166,68,147)(60,158,69,139)(61,150,70,131)(62,163,71,144)(63,155,72,136) );`

`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)], [(1,76,23,46),(2,84,24,54),(3,71,25,62),(4,79,26,49),(5,66,27,57),(6,74,28,44),(7,82,29,52),(8,69,30,60),(9,77,31,47),(10,64,32,55),(11,72,33,63),(12,80,34,50),(13,67,35,58),(14,75,36,45),(15,83,37,53),(16,70,38,61),(17,78,39,48),(18,65,40,56),(19,73,41,43),(20,81,42,51),(21,68,22,59),(85,146,113,165),(86,133,114,152),(87,141,115,160),(88,128,116,168),(89,136,117,155),(90,144,118,163),(91,131,119,150),(92,139,120,158),(93,147,121,166),(94,134,122,153),(95,142,123,161),(96,129,124,148),(97,137,125,156),(98,145,126,164),(99,132,106,151),(100,140,107,159),(101,127,108,167),(102,135,109,154),(103,143,110,162),(104,130,111,149),(105,138,112,157)], [(1,113,23,85),(2,126,24,98),(3,118,25,90),(4,110,26,103),(5,123,27,95),(6,115,28,87),(7,107,29,100),(8,120,30,92),(9,112,31,105),(10,125,32,97),(11,117,33,89),(12,109,34,102),(13,122,35,94),(14,114,36,86),(15,106,37,99),(16,119,38,91),(17,111,39,104),(18,124,40,96),(19,116,41,88),(20,108,42,101),(21,121,22,93),(43,168,73,128),(44,160,74,141),(45,152,75,133),(46,165,76,146),(47,157,77,138),(48,149,78,130),(49,162,79,143),(50,154,80,135),(51,167,81,127),(52,159,82,140),(53,151,83,132),(54,164,84,145),(55,156,64,137),(56,148,65,129),(57,161,66,142),(58,153,67,134),(59,166,68,147),(60,158,69,139),(61,150,70,131),(62,163,71,144),(63,155,72,136)])`

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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