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G = C18×D13order 468 = 22·32·13

Direct product of C18 and D13

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

Aliases: C18×D13, C2342C2, C263C18, C78.11C6, C1173C22, C3.(C6×D13), C133(C2×C18), C39.3(C2×C6), C6.3(C3×D13), (C3×D13).8C6, (C6×D13).2C3, SmallGroup(468,15)

Series: Derived Chief Lower central Upper central

C1C13 — C18×D13
C1C13C39C117C9×D13 — C18×D13
C13 — C18×D13
C1C18

Generators and relations for C18×D13
 G = < a,b,c | a18=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C2
13C22
13C6
13C6
13C2×C6
13C18
13C18
13C2×C18

Smallest permutation representation of C18×D13
On 234 points
Generators in S234
(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 86 208 106 192 117 146 136 36 224 60 171 51)(2 87 209 107 193 118 147 137 19 225 61 172 52)(3 88 210 108 194 119 148 138 20 226 62 173 53)(4 89 211 91 195 120 149 139 21 227 63 174 54)(5 90 212 92 196 121 150 140 22 228 64 175 37)(6 73 213 93 197 122 151 141 23 229 65 176 38)(7 74 214 94 198 123 152 142 24 230 66 177 39)(8 75 215 95 181 124 153 143 25 231 67 178 40)(9 76 216 96 182 125 154 144 26 232 68 179 41)(10 77 199 97 183 126 155 127 27 233 69 180 42)(11 78 200 98 184 109 156 128 28 234 70 163 43)(12 79 201 99 185 110 157 129 29 217 71 164 44)(13 80 202 100 186 111 158 130 30 218 72 165 45)(14 81 203 101 187 112 159 131 31 219 55 166 46)(15 82 204 102 188 113 160 132 32 220 56 167 47)(16 83 205 103 189 114 161 133 33 221 57 168 48)(17 84 206 104 190 115 162 134 34 222 58 169 49)(18 85 207 105 191 116 145 135 35 223 59 170 50)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 37)(15 38)(16 39)(17 40)(18 41)(19 184)(20 185)(21 186)(22 187)(23 188)(24 189)(25 190)(26 191)(27 192)(28 193)(29 194)(30 195)(31 196)(32 197)(33 198)(34 181)(35 182)(36 183)(55 212)(56 213)(57 214)(58 215)(59 216)(60 199)(61 200)(62 201)(63 202)(64 203)(65 204)(66 205)(67 206)(68 207)(69 208)(70 209)(71 210)(72 211)(73 167)(74 168)(75 169)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 177)(84 178)(85 179)(86 180)(87 163)(88 164)(89 165)(90 166)(91 218)(92 219)(93 220)(94 221)(95 222)(96 223)(97 224)(98 225)(99 226)(100 227)(101 228)(102 229)(103 230)(104 231)(105 232)(106 233)(107 234)(108 217)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)(115 143)(116 144)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(145 154)(146 155)(147 156)(148 157)(149 158)(150 159)(151 160)(152 161)(153 162)

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

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

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

144 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F9A···9F13A···13F18A···18F18G···18R26A···26F39A···39L78A···78L117A···117AJ234A···234AJ
order1222336666669···913···1318···1818···1826···2639···3978···78117···117234···234
size1113131111131313131···12···21···113···132···22···22···22···22···2

144 irreducible representations

dim111111111222222
type+++++
imageC1C2C2C3C6C6C9C18C18D13D26C3×D13C6×D13C9×D13C18×D13
kernelC18×D13C9×D13C234C6×D13C3×D13C78D26D13C26C18C9C6C3C2C1
# reps12124261266612123636

Matrix representation of C18×D13 in GL3(𝔽937) generated by

92400
03230
00323
,
100
09041
0361131
,
93600
08061
0643131
G:=sub<GL(3,GF(937))| [924,0,0,0,323,0,0,0,323],[1,0,0,0,904,361,0,1,131],[936,0,0,0,806,643,0,1,131] >;

C18×D13 in GAP, Magma, Sage, TeX

C_{18}\times D_{13}
% in TeX

G:=Group("C18xD13");
// GroupNames label

G:=SmallGroup(468,15);
// by ID

G=gap.SmallGroup(468,15);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,57,10804]);
// Polycyclic

G:=Group<a,b,c|a^18=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C18×D13 in TeX

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