Copied to
clipboard

G = C43⋊D4order 344 = 23·43

The semidirect product of C43 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C432D4, C22⋊D43, D862C2, Dic43⋊C2, C2.5D86, C86.5C22, (C2×C86)⋊2C2, SmallGroup(344,7)

Series: Derived Chief Lower central Upper central

C1C86 — C43⋊D4
C1C43C86D86 — C43⋊D4
C43C86 — C43⋊D4
C1C2C22

Generators and relations for C43⋊D4
 G = < a,b,c | a43=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
86C2
43C4
43C22
2D43
2C86
43D4

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

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

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

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

89 conjugacy classes

class 1 2A2B2C 4 43A···43U86A···86BK
order1222443···4386···86
size11286862···22···2

89 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D43D86C43⋊D4
kernelC43⋊D4Dic43D86C2×C86C43C22C2C1
# reps11111212142

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

1571
15755
,
126146
3747
,
1098
29163
G:=sub<GL(2,GF(173))| [157,157,1,55],[126,37,146,47],[10,29,98,163] >;

C43⋊D4 in GAP, Magma, Sage, TeX

C_{43}\rtimes D_4
% in TeX

G:=Group("C43:D4");
// GroupNames label

G:=SmallGroup(344,7);
// by ID

G=gap.SmallGroup(344,7);
# by ID

G:=PCGroup([4,-2,-2,-2,-43,49,5379]);
// Polycyclic

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

Export

Subgroup lattice of C43⋊D4 in TeX

׿
×
𝔽