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G = C4×D43order 344 = 23·43

Direct product of C4 and D43

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D43, D86.C2, C1722C2, C2.1D86, Dic432C2, C86.2C22, C431(C2×C4), SmallGroup(344,4)

Series: Derived Chief Lower central Upper central

C1C43 — C4×D43
C1C43C86D86 — C4×D43
C43 — C4×D43
C1C4

Generators and relations for C4×D43
 G = < a,b,c | a4=b43=c2=1, ab=ba, ac=ca, cbc=b-1 >

43C2
43C2
43C22
43C4
43C2×C4

Smallest permutation representation of C4×D43
On 172 points
Generators in S172
(1 150 81 99)(2 151 82 100)(3 152 83 101)(4 153 84 102)(5 154 85 103)(6 155 86 104)(7 156 44 105)(8 157 45 106)(9 158 46 107)(10 159 47 108)(11 160 48 109)(12 161 49 110)(13 162 50 111)(14 163 51 112)(15 164 52 113)(16 165 53 114)(17 166 54 115)(18 167 55 116)(19 168 56 117)(20 169 57 118)(21 170 58 119)(22 171 59 120)(23 172 60 121)(24 130 61 122)(25 131 62 123)(26 132 63 124)(27 133 64 125)(28 134 65 126)(29 135 66 127)(30 136 67 128)(31 137 68 129)(32 138 69 87)(33 139 70 88)(34 140 71 89)(35 141 72 90)(36 142 73 91)(37 143 74 92)(38 144 75 93)(39 145 76 94)(40 146 77 95)(41 147 78 96)(42 148 79 97)(43 149 80 98)
(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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(44 74)(45 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(130 169)(131 168)(132 167)(133 166)(134 165)(135 164)(136 163)(137 162)(138 161)(139 160)(140 159)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(170 172)

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

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

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

92 conjugacy classes

class 1 2A2B2C4A4B4C4D43A···43U86A···86U172A···172AP
order1222444443···4386···86172···172
size1143431143432···22···22···2

92 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D43D86C4×D43
kernelC4×D43Dic43C172D86D43C4C2C1
# reps11114212142

Matrix representation of C4×D43 in GL3(𝔽173) generated by

9300
01720
00172
,
100
001
017211
,
17200
001
010
G:=sub<GL(3,GF(173))| [93,0,0,0,172,0,0,0,172],[1,0,0,0,0,172,0,1,11],[172,0,0,0,0,1,0,1,0] >;

C4×D43 in GAP, Magma, Sage, TeX

C_4\times D_{43}
% in TeX

G:=Group("C4xD43");
// GroupNames label

G:=SmallGroup(344,4);
// by ID

G=gap.SmallGroup(344,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-43,21,5379]);
// Polycyclic

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

Export

Subgroup lattice of C4×D43 in TeX

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