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G = C21⋊Q8order 168 = 23·3·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C42 — C21⋊Q8
C1C7C21C42C3×Dic7 — C21⋊Q8
C21C42 — C21⋊Q8
C1C2

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 >

3C4
7C4
21C4
21Q8
7C12
7Dic3
3C28
3Dic7
7Dic6
3Dic14

Character table of C21⋊Q8

 class 1234A4B4C67A7B7C12A12B14A14B14C21A21B21C28A28B28C28D28E28F42A42B42C
 size 1126144222221414222444666666444
ρ1111111111111111111111111111    trivial
ρ2111-1-111111-1-1111111-1-1-1-1-1-1111    linear of order 2
ρ31111-1-11111-1-1111111111111111    linear of order 2
ρ4111-11-1111111111111-1-1-1-1-1-1111    linear of order 2
ρ522-10-20-122211222-1-1-1000000-1-1-1    orthogonal lifted from D6
ρ6222-2002ζ767ζ7473ζ757200ζ767ζ7572ζ7473ζ767ζ7572ζ74737677572757276774737473ζ7572ζ7473ζ767    orthogonal lifted from D14
ρ722-1020-1222-1-1222-1-1-1000000-1-1-1    orthogonal lifted from S3
ρ82222002ζ7473ζ7572ζ76700ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ767ζ7473ζ7572ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ9222-2002ζ7572ζ767ζ747300ζ7572ζ7473ζ767ζ7572ζ7473ζ7677572747374737572767767ζ7473ζ767ζ7572    orthogonal lifted from D14
ρ102222002ζ7572ζ767ζ747300ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ7473ζ7572ζ767ζ767ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ112222002ζ767ζ7473ζ757200ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7572ζ767ζ7473ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ12222-2002ζ7473ζ7572ζ76700ζ7473ζ767ζ7572ζ7473ζ767ζ75727473767767747375727572ζ767ζ7572ζ7473    orthogonal lifted from D14
ρ132-22000-222200-2-2-2222000000-2-2-2    symplectic lifted from Q8, Schur index 2
ρ142-2-10001222-33-2-2-2-1-1-1000000111    symplectic lifted from Dic6, Schur index 2
ρ152-2-100012223-3-2-2-2-1-1-1000000111    symplectic lifted from Dic6, Schur index 2
ρ162-22000-2ζ767ζ7473ζ75720076775727473ζ767ζ7572ζ74734ζ764ζ743ζ7543ζ72ζ43ζ7543ζ72ζ4ζ764ζ74ζ744ζ73ζ4ζ744ζ7375727473767    symplectic lifted from Dic14, Schur index 2
ρ172-22000-2ζ7473ζ7572ζ7670074737677572ζ7473ζ767ζ7572ζ4ζ744ζ73ζ4ζ764ζ74ζ764ζ74ζ744ζ7343ζ7543ζ72ζ43ζ7543ζ7276775727473    symplectic lifted from Dic14, Schur index 2
ρ182-22000-2ζ7473ζ7572ζ7670074737677572ζ7473ζ767ζ75724ζ744ζ734ζ764ζ7ζ4ζ764ζ7ζ4ζ744ζ73ζ43ζ7543ζ7243ζ7543ζ7276775727473    symplectic lifted from Dic14, Schur index 2
ρ192-22000-2ζ767ζ7473ζ75720076775727473ζ767ζ7572ζ7473ζ4ζ764ζ7ζ43ζ7543ζ7243ζ7543ζ724ζ764ζ7ζ4ζ744ζ734ζ744ζ7375727473767    symplectic lifted from Dic14, Schur index 2
ρ202-22000-2ζ7572ζ767ζ74730075727473767ζ7572ζ7473ζ767ζ43ζ7543ζ724ζ744ζ73ζ4ζ744ζ7343ζ7543ζ72ζ4ζ764ζ74ζ764ζ774737677572    symplectic lifted from Dic14, Schur index 2
ρ212-22000-2ζ7572ζ767ζ74730075727473767ζ7572ζ7473ζ76743ζ7543ζ72ζ4ζ744ζ734ζ744ζ73ζ43ζ7543ζ724ζ764ζ7ζ4ζ764ζ774737677572    symplectic lifted from Dic14, Schur index 2
ρ2244-2000-276+2ζ774+2ζ7375+2ζ720076+2ζ775+2ζ7274+2ζ737677572747300000075727473767    orthogonal lifted from S3×D7
ρ2344-2000-274+2ζ7375+2ζ7276+2ζ70074+2ζ7376+2ζ775+2ζ727473767757200000076775727473    orthogonal lifted from S3×D7
ρ2444-2000-275+2ζ7276+2ζ774+2ζ730075+2ζ7274+2ζ7376+2ζ77572747376700000074737677572    orthogonal lifted from S3×D7
ρ254-4-2000276+2ζ774+2ζ7375+2ζ7200-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ7376775727473000000ζ7572ζ7473ζ767    symplectic faithful, Schur index 2
ρ264-4-2000275+2ζ7276+2ζ774+2ζ7300-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ775727473767000000ζ7473ζ767ζ7572    symplectic faithful, Schur index 2
ρ274-4-2000274+2ζ7375+2ζ7276+2ζ700-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ7274737677572000000ζ767ζ7572ζ7473    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

22711000
1177600
0001
00336336
,
1000
0100
00142218
0076195
,
8311500
11325400
00322307
003015
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

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