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G = C2×C13⋊C9order 234 = 2·32·13

Direct product of C2 and C13⋊C9

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C2×C13⋊C9, C26⋊C9, C78.C3, C132C18, C39.2C6, C6.(C13⋊C3), C3.(C2×C13⋊C3), SmallGroup(234,2)

Series: Derived Chief Lower central Upper central

C1C13 — C2×C13⋊C9
C1C13C39C13⋊C9 — C2×C13⋊C9
C13 — C2×C13⋊C9
C1C6

Generators and relations for C2×C13⋊C9
 G = < a,b,c | a2=b13=c9=1, ab=ba, ac=ca, cbc-1=b9 >

13C9
13C18

Smallest permutation representation of C2×C13⋊C9
Regular action on 234 points
Generators in S234
(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 138)(22 139)(23 140)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 165)(41 166)(42 167)(43 168)(44 169)(45 157)(46 158)(47 159)(48 160)(49 161)(50 162)(51 163)(52 164)(53 172)(54 173)(55 174)(56 175)(57 176)(58 177)(59 178)(60 179)(61 180)(62 181)(63 182)(64 170)(65 171)(66 185)(67 186)(68 187)(69 188)(70 189)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 183)(78 184)(79 208)(80 196)(81 197)(82 198)(83 199)(84 200)(85 201)(86 202)(87 203)(88 204)(89 205)(90 206)(91 207)(92 216)(93 217)(94 218)(95 219)(96 220)(97 221)(98 209)(99 210)(100 211)(101 212)(102 213)(103 214)(104 215)(105 229)(106 230)(107 231)(108 232)(109 233)(110 234)(111 222)(112 223)(113 224)(114 225)(115 226)(116 227)(117 228)
(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)(170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195)(196 197 198 199 200 201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216 217 218 219 220 221)(222 223 224 225 226 227 228 229 230 231 232 233 234)
(1 114 72 27 96 62 14 84 48)(2 117 68 28 99 58 15 87 44)(3 107 77 29 102 54 16 90 40)(4 110 73 30 92 63 17 80 49)(5 113 69 31 95 59 18 83 45)(6 116 78 32 98 55 19 86 41)(7 106 74 33 101 64 20 89 50)(8 109 70 34 104 60 21 79 46)(9 112 66 35 94 56 22 82 42)(10 115 75 36 97 65 23 85 51)(11 105 71 37 100 61 24 88 47)(12 108 67 38 103 57 25 91 43)(13 111 76 39 93 53 26 81 52)(118 225 191 144 220 181 131 200 160)(119 228 187 145 210 177 132 203 169)(120 231 183 146 213 173 133 206 165)(121 234 192 147 216 182 134 196 161)(122 224 188 148 219 178 135 199 157)(123 227 184 149 209 174 136 202 166)(124 230 193 150 212 170 137 205 162)(125 233 189 151 215 179 138 208 158)(126 223 185 152 218 175 139 198 167)(127 226 194 153 221 171 140 201 163)(128 229 190 154 211 180 141 204 159)(129 232 186 155 214 176 142 207 168)(130 222 195 156 217 172 143 197 164)

G:=sub<Sym(234)| (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,138)(22,139)(23,140)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,165)(41,166)(42,167)(43,168)(44,169)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,172)(54,173)(55,174)(56,175)(57,176)(58,177)(59,178)(60,179)(61,180)(62,181)(63,182)(64,170)(65,171)(66,185)(67,186)(68,187)(69,188)(70,189)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,183)(78,184)(79,208)(80,196)(81,197)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,216)(93,217)(94,218)(95,219)(96,220)(97,221)(98,209)(99,210)(100,211)(101,212)(102,213)(103,214)(104,215)(105,229)(106,230)(107,231)(108,232)(109,233)(110,234)(111,222)(112,223)(113,224)(114,225)(115,226)(116,227)(117,228), (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)(170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221)(222,223,224,225,226,227,228,229,230,231,232,233,234), (1,114,72,27,96,62,14,84,48)(2,117,68,28,99,58,15,87,44)(3,107,77,29,102,54,16,90,40)(4,110,73,30,92,63,17,80,49)(5,113,69,31,95,59,18,83,45)(6,116,78,32,98,55,19,86,41)(7,106,74,33,101,64,20,89,50)(8,109,70,34,104,60,21,79,46)(9,112,66,35,94,56,22,82,42)(10,115,75,36,97,65,23,85,51)(11,105,71,37,100,61,24,88,47)(12,108,67,38,103,57,25,91,43)(13,111,76,39,93,53,26,81,52)(118,225,191,144,220,181,131,200,160)(119,228,187,145,210,177,132,203,169)(120,231,183,146,213,173,133,206,165)(121,234,192,147,216,182,134,196,161)(122,224,188,148,219,178,135,199,157)(123,227,184,149,209,174,136,202,166)(124,230,193,150,212,170,137,205,162)(125,233,189,151,215,179,138,208,158)(126,223,185,152,218,175,139,198,167)(127,226,194,153,221,171,140,201,163)(128,229,190,154,211,180,141,204,159)(129,232,186,155,214,176,142,207,168)(130,222,195,156,217,172,143,197,164)>;

G:=Group( (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,138)(22,139)(23,140)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,165)(41,166)(42,167)(43,168)(44,169)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,172)(54,173)(55,174)(56,175)(57,176)(58,177)(59,178)(60,179)(61,180)(62,181)(63,182)(64,170)(65,171)(66,185)(67,186)(68,187)(69,188)(70,189)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,183)(78,184)(79,208)(80,196)(81,197)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,216)(93,217)(94,218)(95,219)(96,220)(97,221)(98,209)(99,210)(100,211)(101,212)(102,213)(103,214)(104,215)(105,229)(106,230)(107,231)(108,232)(109,233)(110,234)(111,222)(112,223)(113,224)(114,225)(115,226)(116,227)(117,228), (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)(170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221)(222,223,224,225,226,227,228,229,230,231,232,233,234), (1,114,72,27,96,62,14,84,48)(2,117,68,28,99,58,15,87,44)(3,107,77,29,102,54,16,90,40)(4,110,73,30,92,63,17,80,49)(5,113,69,31,95,59,18,83,45)(6,116,78,32,98,55,19,86,41)(7,106,74,33,101,64,20,89,50)(8,109,70,34,104,60,21,79,46)(9,112,66,35,94,56,22,82,42)(10,115,75,36,97,65,23,85,51)(11,105,71,37,100,61,24,88,47)(12,108,67,38,103,57,25,91,43)(13,111,76,39,93,53,26,81,52)(118,225,191,144,220,181,131,200,160)(119,228,187,145,210,177,132,203,169)(120,231,183,146,213,173,133,206,165)(121,234,192,147,216,182,134,196,161)(122,224,188,148,219,178,135,199,157)(123,227,184,149,209,174,136,202,166)(124,230,193,150,212,170,137,205,162)(125,233,189,151,215,179,138,208,158)(126,223,185,152,218,175,139,198,167)(127,226,194,153,221,171,140,201,163)(128,229,190,154,211,180,141,204,159)(129,232,186,155,214,176,142,207,168)(130,222,195,156,217,172,143,197,164) );

G=PermutationGroup([(1,118),(2,119),(3,120),(4,121),(5,122),(6,123),(7,124),(8,125),(9,126),(10,127),(11,128),(12,129),(13,130),(14,131),(15,132),(16,133),(17,134),(18,135),(19,136),(20,137),(21,138),(22,139),(23,140),(24,141),(25,142),(26,143),(27,144),(28,145),(29,146),(30,147),(31,148),(32,149),(33,150),(34,151),(35,152),(36,153),(37,154),(38,155),(39,156),(40,165),(41,166),(42,167),(43,168),(44,169),(45,157),(46,158),(47,159),(48,160),(49,161),(50,162),(51,163),(52,164),(53,172),(54,173),(55,174),(56,175),(57,176),(58,177),(59,178),(60,179),(61,180),(62,181),(63,182),(64,170),(65,171),(66,185),(67,186),(68,187),(69,188),(70,189),(71,190),(72,191),(73,192),(74,193),(75,194),(76,195),(77,183),(78,184),(79,208),(80,196),(81,197),(82,198),(83,199),(84,200),(85,201),(86,202),(87,203),(88,204),(89,205),(90,206),(91,207),(92,216),(93,217),(94,218),(95,219),(96,220),(97,221),(98,209),(99,210),(100,211),(101,212),(102,213),(103,214),(104,215),(105,229),(106,230),(107,231),(108,232),(109,233),(110,234),(111,222),(112,223),(113,224),(114,225),(115,226),(116,227),(117,228)], [(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),(170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195),(196,197,198,199,200,201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216,217,218,219,220,221),(222,223,224,225,226,227,228,229,230,231,232,233,234)], [(1,114,72,27,96,62,14,84,48),(2,117,68,28,99,58,15,87,44),(3,107,77,29,102,54,16,90,40),(4,110,73,30,92,63,17,80,49),(5,113,69,31,95,59,18,83,45),(6,116,78,32,98,55,19,86,41),(7,106,74,33,101,64,20,89,50),(8,109,70,34,104,60,21,79,46),(9,112,66,35,94,56,22,82,42),(10,115,75,36,97,65,23,85,51),(11,105,71,37,100,61,24,88,47),(12,108,67,38,103,57,25,91,43),(13,111,76,39,93,53,26,81,52),(118,225,191,144,220,181,131,200,160),(119,228,187,145,210,177,132,203,169),(120,231,183,146,213,173,133,206,165),(121,234,192,147,216,182,134,196,161),(122,224,188,148,219,178,135,199,157),(123,227,184,149,209,174,136,202,166),(124,230,193,150,212,170,137,205,162),(125,233,189,151,215,179,138,208,158),(126,223,185,152,218,175,139,198,167),(127,226,194,153,221,171,140,201,163),(128,229,190,154,211,180,141,204,159),(129,232,186,155,214,176,142,207,168),(130,222,195,156,217,172,143,197,164)])

C2×C13⋊C9 is a maximal subgroup of   C132C36

42 conjugacy classes

class 1  2 3A3B6A6B9A···9F13A13B13C13D18A···18F26A26B26C26D39A···39H78A···78H
order1233669···91313131318···182626262639···3978···78
size11111113···13333313···1333333···33···3

42 irreducible representations

dim1111113333
type++
imageC1C2C3C6C9C18C13⋊C3C2×C13⋊C3C13⋊C9C2×C13⋊C9
kernelC2×C13⋊C9C13⋊C9C78C39C26C13C6C3C2C1
# reps1122664488

Matrix representation of C2×C13⋊C9 in GL4(𝔽937) generated by

936000
093600
009360
000936
,
1000
06614821
06624821
06614831
,
169000
0857171187
0588771646
0887392246
G:=sub<GL(4,GF(937))| [936,0,0,0,0,936,0,0,0,0,936,0,0,0,0,936],[1,0,0,0,0,661,662,661,0,482,482,483,0,1,1,1],[169,0,0,0,0,857,588,887,0,171,771,392,0,187,646,246] >;

C2×C13⋊C9 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes C_9
% in TeX

G:=Group("C2xC13:C9");
// GroupNames label

G:=SmallGroup(234,2);
// by ID

G=gap.SmallGroup(234,2);
# by ID

G:=PCGroup([4,-2,-3,-3,-13,29,439]);
// Polycyclic

G:=Group<a,b,c|a^2=b^13=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

Export

Subgroup lattice of C2×C13⋊C9 in TeX

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