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## G = D8×C21order 336 = 24·3·7

### Direct product of C21 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: D8×C21, D4⋊C42, C81C42, C5613C6, C243C14, C16811C2, C42.54D4, C84.77C22, (C7×D4)⋊10C6, (C3×D4)⋊4C14, C4.1(C2×C42), C2.3(D4×C21), C6.14(C7×D4), (D4×C21)⋊10C2, C28.40(C2×C6), C14.30(C3×D4), C12.17(C2×C14), SmallGroup(336,111)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C21
 Chief series C1 — C2 — C4 — C28 — C84 — D4×C21 — D8×C21
 Lower central C1 — C2 — C4 — D8×C21
 Upper central C1 — C42 — C84 — D8×C21

Generators and relations for D8×C21
G = < a,b,c | a21=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D8×C21
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 28 83 152 43 140 107 96)(2 29 84 153 44 141 108 97)(3 30 64 154 45 142 109 98)(4 31 65 155 46 143 110 99)(5 32 66 156 47 144 111 100)(6 33 67 157 48 145 112 101)(7 34 68 158 49 146 113 102)(8 35 69 159 50 147 114 103)(9 36 70 160 51 127 115 104)(10 37 71 161 52 128 116 105)(11 38 72 162 53 129 117 85)(12 39 73 163 54 130 118 86)(13 40 74 164 55 131 119 87)(14 41 75 165 56 132 120 88)(15 42 76 166 57 133 121 89)(16 22 77 167 58 134 122 90)(17 23 78 168 59 135 123 91)(18 24 79 148 60 136 124 92)(19 25 80 149 61 137 125 93)(20 26 81 150 62 138 126 94)(21 27 82 151 63 139 106 95)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 122)(23 123)(24 124)(25 125)(26 126)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 121)(43 152)(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 161)(53 162)(54 163)(55 164)(56 165)(57 166)(58 167)(59 168)(60 148)(61 149)(62 150)(63 151)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 137)(81 138)(82 139)(83 140)(84 141)

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

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

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

147 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 7A ··· 7F 8A 8B 12A 12B 14A ··· 14F 14G ··· 14R 21A ··· 21L 24A 24B 24C 24D 28A ··· 28F 42A ··· 42L 42M ··· 42AJ 56A ··· 56L 84A ··· 84L 168A ··· 168X order 1 2 2 2 3 3 4 6 6 6 6 6 6 7 ··· 7 8 8 12 12 14 ··· 14 14 ··· 14 21 ··· 21 24 24 24 24 28 ··· 28 42 ··· 42 42 ··· 42 56 ··· 56 84 ··· 84 168 ··· 168 size 1 1 4 4 1 1 2 1 1 4 4 4 4 1 ··· 1 2 2 2 2 1 ··· 1 4 ··· 4 1 ··· 1 2 2 2 2 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2 2 ··· 2

147 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 C7 C14 C14 C21 C42 C42 D4 D8 C3×D4 C3×D8 C7×D4 C7×D8 D4×C21 D8×C21 kernel D8×C21 C168 D4×C21 C7×D8 C56 C7×D4 C3×D8 C24 C3×D4 D8 C8 D4 C42 C21 C14 C7 C6 C3 C2 C1 # reps 1 1 2 2 2 4 6 6 12 12 12 24 1 2 2 4 6 12 12 24

Matrix representation of D8×C21 in GL2(𝔽337) generated by

 4 0 0 4
,
 324 13 324 324
,
 324 13 13 13
G:=sub<GL(2,GF(337))| [4,0,0,4],[324,324,13,324],[324,13,13,13] >;

D8×C21 in GAP, Magma, Sage, TeX

D_8\times C_{21}
% in TeX

G:=Group("D8xC21");
// GroupNames label

G:=SmallGroup(336,111);
// by ID

G=gap.SmallGroup(336,111);
# by ID

G:=PCGroup([6,-2,-2,-3,-7,-2,-2,1033,7564,3790,88]);
// Polycyclic

G:=Group<a,b,c|a^21=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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