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## G = C2×Dic21order 168 = 23·3·7

### Direct product of C2 and Dic21

Aliases: C2×Dic21, C6⋊Dic7, C421C4, C14⋊Dic3, C2.2D42, C22.D21, C14.11D6, C6.11D14, C42.11C22, (C2×C6).D7, C217(C2×C4), (C2×C14).S3, C72(C2×Dic3), C32(C2×Dic7), (C2×C42).1C2, SmallGroup(168,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C2×Dic21
 Chief series C1 — C7 — C21 — C42 — Dic21 — C2×Dic21
 Lower central C21 — C2×Dic21
 Upper central C1 — C22

Generators and relations for C2×Dic21
G = < a,b,c | a2=b42=1, c2=b21, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C2×Dic21
Regular action on 168 points
Generators in S168
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 157)(102 158)(103 159)(104 160)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)
(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)
(1 141 22 162)(2 140 23 161)(3 139 24 160)(4 138 25 159)(5 137 26 158)(6 136 27 157)(7 135 28 156)(8 134 29 155)(9 133 30 154)(10 132 31 153)(11 131 32 152)(12 130 33 151)(13 129 34 150)(14 128 35 149)(15 127 36 148)(16 168 37 147)(17 167 38 146)(18 166 39 145)(19 165 40 144)(20 164 41 143)(21 163 42 142)(43 85 64 106)(44 126 65 105)(45 125 66 104)(46 124 67 103)(47 123 68 102)(48 122 69 101)(49 121 70 100)(50 120 71 99)(51 119 72 98)(52 118 73 97)(53 117 74 96)(54 116 75 95)(55 115 76 94)(56 114 77 93)(57 113 78 92)(58 112 79 91)(59 111 80 90)(60 110 81 89)(61 109 82 88)(62 108 83 87)(63 107 84 86)

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

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

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

C2×Dic21 is a maximal subgroup of
Dic3×Dic7  D14⋊Dic3  D6⋊Dic7  C42.Q8  Dic21⋊C4  C14.Dic6  C42.4Q8  C84⋊C4  C2.D84  C42.38D4  C2×Dic3×D7  C42.C23  C2×S3×Dic7  C2×C4×D21  D42D21
C2×Dic21 is a maximal quotient of
C84.C4  C84⋊C4  C42.38D4

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 7A 7B 7C 14A ··· 14I 21A ··· 21F 42A ··· 42R order 1 2 2 2 3 4 4 4 4 6 6 6 7 7 7 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 1 1 2 21 21 21 21 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + + - + + - + image C1 C2 C2 C4 S3 Dic3 D6 D7 Dic7 D14 D21 Dic21 D42 kernel C2×Dic21 Dic21 C2×C42 C42 C2×C14 C14 C14 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 4 1 2 1 3 6 3 6 12 6

Matrix representation of C2×Dic21 in GL5(𝔽337)

 1 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 1 0 0 0 0 0 1
,
 336 0 0 0 0 0 0 1 0 0 0 336 227 0 0 0 0 0 5 236 0 0 0 202 166
,
 148 0 0 0 0 0 68 331 0 0 0 265 269 0 0 0 0 0 243 251 0 0 0 138 94

G:=sub<GL(5,GF(337))| [1,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1],[336,0,0,0,0,0,0,336,0,0,0,1,227,0,0,0,0,0,5,202,0,0,0,236,166],[148,0,0,0,0,0,68,265,0,0,0,331,269,0,0,0,0,0,243,138,0,0,0,251,94] >;

C2×Dic21 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{21}
% in TeX

G:=Group("C2xDic21");
// GroupNames label

G:=SmallGroup(168,37);
// by ID

G=gap.SmallGroup(168,37);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,20,323,3604]);
// Polycyclic

G:=Group<a,b,c|a^2=b^42=1,c^2=b^21,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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