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G = C22×C42order 168 = 23·3·7

Abelian group of type [2,2,42]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C42, SmallGroup(168,57)

Series: Derived Chief Lower central Upper central

C1 — C22×C42
C1C7C21C42C2×C42 — C22×C42
C1 — C22×C42
C1 — C22×C42

Generators and relations for C22×C42
 G = < a,b,c | a2=b2=c42=1, ab=ba, ac=ca, bc=cb >

Subgroups: 64, all normal (8 characteristic)
C1, C2 [×7], C3, C22 [×7], C6 [×7], C7, C23, C2×C6 [×7], C14 [×7], C21, C22×C6, C2×C14 [×7], C42 [×7], C22×C14, C2×C42 [×7], C22×C42
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C7, C23, C2×C6 [×7], C14 [×7], C21, C22×C6, C2×C14 [×7], C42 [×7], C22×C14, C2×C42 [×7], C22×C42

Smallest permutation representation of C22×C42
Regular action on 168 points
Generators in S168
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(49 142)(50 143)(51 144)(52 145)(53 146)(54 147)(55 148)(56 149)(57 150)(58 151)(59 152)(60 153)(61 154)(62 155)(63 156)(64 157)(65 158)(66 159)(67 160)(68 161)(69 162)(70 163)(71 164)(72 165)(73 166)(74 167)(75 168)(76 127)(77 128)(78 129)(79 130)(80 131)(81 132)(82 133)(83 134)(84 135)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 50)(85 164)(86 165)(87 166)(88 167)(89 168)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)(101 138)(102 139)(103 140)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)(121 158)(122 159)(123 160)(124 161)(125 162)(126 163)
(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)

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

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

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

C22×C42 is a maximal subgroup of   C42.38D4

168 conjugacy classes

class 1 2A···2G3A3B6A···6N7A···7F14A···14AP21A···21L42A···42CF
order12···2336···67···714···1421···2142···42
size11···1111···11···11···11···11···1

168 irreducible representations

dim11111111
type++
imageC1C2C3C6C7C14C21C42
kernelC22×C42C2×C42C22×C14C2×C14C22×C6C2×C6C23C22
# reps172146421284

Matrix representation of C22×C42 in GL3(𝔽43) generated by

4200
010
001
,
100
010
0042
,
1000
0330
005
G:=sub<GL(3,GF(43))| [42,0,0,0,1,0,0,0,1],[1,0,0,0,1,0,0,0,42],[10,0,0,0,33,0,0,0,5] >;

C22×C42 in GAP, Magma, Sage, TeX

C_2^2\times C_{42}
% in TeX

G:=Group("C2^2xC42");
// GroupNames label

G:=SmallGroup(168,57);
// by ID

G=gap.SmallGroup(168,57);
# by ID

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

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

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