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G = D4×C43order 344 = 23·43

Direct product of C43 and D4

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

Aliases: D4×C43, C4⋊C86, C22⋊C86, C1723C2, C86.6C22, (C2×C86)⋊1C2, C2.1(C2×C86), SmallGroup(344,9)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C43
C1C2C86C2×C86 — D4×C43
C1C2 — D4×C43
C1C86 — D4×C43

Generators and relations for D4×C43
 G = < a,b,c | a43=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C86
2C86

Smallest permutation representation of D4×C43
On 172 points
Generators in S172
(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)(130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172)
(1 155 104 66)(2 156 105 67)(3 157 106 68)(4 158 107 69)(5 159 108 70)(6 160 109 71)(7 161 110 72)(8 162 111 73)(9 163 112 74)(10 164 113 75)(11 165 114 76)(12 166 115 77)(13 167 116 78)(14 168 117 79)(15 169 118 80)(16 170 119 81)(17 171 120 82)(18 172 121 83)(19 130 122 84)(20 131 123 85)(21 132 124 86)(22 133 125 44)(23 134 126 45)(24 135 127 46)(25 136 128 47)(26 137 129 48)(27 138 87 49)(28 139 88 50)(29 140 89 51)(30 141 90 52)(31 142 91 53)(32 143 92 54)(33 144 93 55)(34 145 94 56)(35 146 95 57)(36 147 96 58)(37 148 97 59)(38 149 98 60)(39 150 99 61)(40 151 100 62)(41 152 101 63)(42 153 102 64)(43 154 103 65)
(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 161)(73 162)(74 163)(75 164)(76 165)(77 166)(78 167)(79 168)(80 169)(81 170)(82 171)(83 172)(84 130)(85 131)(86 132)

G:=sub<Sym(172)| (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)(130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,155,104,66)(2,156,105,67)(3,157,106,68)(4,158,107,69)(5,159,108,70)(6,160,109,71)(7,161,110,72)(8,162,111,73)(9,163,112,74)(10,164,113,75)(11,165,114,76)(12,166,115,77)(13,167,116,78)(14,168,117,79)(15,169,118,80)(16,170,119,81)(17,171,120,82)(18,172,121,83)(19,130,122,84)(20,131,123,85)(21,132,124,86)(22,133,125,44)(23,134,126,45)(24,135,127,46)(25,136,128,47)(26,137,129,48)(27,138,87,49)(28,139,88,50)(29,140,89,51)(30,141,90,52)(31,142,91,53)(32,143,92,54)(33,144,93,55)(34,145,94,56)(35,146,95,57)(36,147,96,58)(37,148,97,59)(38,149,98,60)(39,150,99,61)(40,151,100,62)(41,152,101,63)(42,153,102,64)(43,154,103,65), (44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,161)(73,162)(74,163)(75,164)(76,165)(77,166)(78,167)(79,168)(80,169)(81,170)(82,171)(83,172)(84,130)(85,131)(86,132)>;

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)(130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,155,104,66)(2,156,105,67)(3,157,106,68)(4,158,107,69)(5,159,108,70)(6,160,109,71)(7,161,110,72)(8,162,111,73)(9,163,112,74)(10,164,113,75)(11,165,114,76)(12,166,115,77)(13,167,116,78)(14,168,117,79)(15,169,118,80)(16,170,119,81)(17,171,120,82)(18,172,121,83)(19,130,122,84)(20,131,123,85)(21,132,124,86)(22,133,125,44)(23,134,126,45)(24,135,127,46)(25,136,128,47)(26,137,129,48)(27,138,87,49)(28,139,88,50)(29,140,89,51)(30,141,90,52)(31,142,91,53)(32,143,92,54)(33,144,93,55)(34,145,94,56)(35,146,95,57)(36,147,96,58)(37,148,97,59)(38,149,98,60)(39,150,99,61)(40,151,100,62)(41,152,101,63)(42,153,102,64)(43,154,103,65), (44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,161)(73,162)(74,163)(75,164)(76,165)(77,166)(78,167)(79,168)(80,169)(81,170)(82,171)(83,172)(84,130)(85,131)(86,132) );

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),(130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172)], [(1,155,104,66),(2,156,105,67),(3,157,106,68),(4,158,107,69),(5,159,108,70),(6,160,109,71),(7,161,110,72),(8,162,111,73),(9,163,112,74),(10,164,113,75),(11,165,114,76),(12,166,115,77),(13,167,116,78),(14,168,117,79),(15,169,118,80),(16,170,119,81),(17,171,120,82),(18,172,121,83),(19,130,122,84),(20,131,123,85),(21,132,124,86),(22,133,125,44),(23,134,126,45),(24,135,127,46),(25,136,128,47),(26,137,129,48),(27,138,87,49),(28,139,88,50),(29,140,89,51),(30,141,90,52),(31,142,91,53),(32,143,92,54),(33,144,93,55),(34,145,94,56),(35,146,95,57),(36,147,96,58),(37,148,97,59),(38,149,98,60),(39,150,99,61),(40,151,100,62),(41,152,101,63),(42,153,102,64),(43,154,103,65)], [(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,140),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,150),(62,151),(63,152),(64,153),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,161),(73,162),(74,163),(75,164),(76,165),(77,166),(78,167),(79,168),(80,169),(81,170),(82,171),(83,172),(84,130),(85,131),(86,132)])

215 conjugacy classes

class 1 2A2B2C 4 43A···43AP86A···86AP86AQ···86DV172A···172AP
order1222443···4386···8686···86172···172
size112221···11···12···22···2

215 irreducible representations

dim11111122
type++++
imageC1C2C2C43C86C86D4D4×C43
kernelD4×C43C172C2×C86D4C4C22C43C1
# reps112424284142

Matrix representation of D4×C43 in GL2(𝔽173) generated by

1170
0117
,
11
171172
,
11
0172
G:=sub<GL(2,GF(173))| [117,0,0,117],[1,171,1,172],[1,0,1,172] >;

D4×C43 in GAP, Magma, Sage, TeX

D_4\times C_{43}
% in TeX

G:=Group("D4xC43");
// GroupNames label

G:=SmallGroup(344,9);
// by ID

G=gap.SmallGroup(344,9);
# by ID

G:=PCGroup([4,-2,-2,-43,-2,1393]);
// Polycyclic

G:=Group<a,b,c|a^43=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C43 in TeX

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