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G = C13×Dic5order 260 = 22·5·13

Direct product of C13 and Dic5

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

Aliases: C13×Dic5, C52C52, C659C4, C10.C26, C26.2D5, C130.3C2, C2.(D5×C13), SmallGroup(260,1)

Series: Derived Chief Lower central Upper central

C1C5 — C13×Dic5
C1C5C10C130 — C13×Dic5
C5 — C13×Dic5
C1C26

Generators and relations for C13×Dic5
 G = < a,b,c | a13=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

5C4
5C52

Smallest permutation representation of C13×Dic5
Regular action on 260 points
Generators in S260
(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)(235 236 237 238 239 240 241 242 243 244 245 246 247)(248 249 250 251 252 253 254 255 256 257 258 259 260)
(1 93 63 17 225 194 130 160 27 204)(2 94 64 18 226 195 118 161 28 205)(3 95 65 19 227 183 119 162 29 206)(4 96 53 20 228 184 120 163 30 207)(5 97 54 21 229 185 121 164 31 208)(6 98 55 22 230 186 122 165 32 196)(7 99 56 23 231 187 123 166 33 197)(8 100 57 24 232 188 124 167 34 198)(9 101 58 25 233 189 125 168 35 199)(10 102 59 26 234 190 126 169 36 200)(11 103 60 14 222 191 127 157 37 201)(12 104 61 15 223 192 128 158 38 202)(13 92 62 16 224 193 129 159 39 203)(40 178 241 80 214 132 150 108 75 251)(41 179 242 81 215 133 151 109 76 252)(42 180 243 82 216 134 152 110 77 253)(43 181 244 83 217 135 153 111 78 254)(44 182 245 84 218 136 154 112 66 255)(45 170 246 85 219 137 155 113 67 256)(46 171 247 86 220 138 156 114 68 257)(47 172 235 87 221 139 144 115 69 258)(48 173 236 88 209 140 145 116 70 259)(49 174 237 89 210 141 146 117 71 260)(50 175 238 90 211 142 147 105 72 248)(51 176 239 91 212 143 148 106 73 249)(52 177 240 79 213 131 149 107 74 250)
(1 236 194 116)(2 237 195 117)(3 238 183 105)(4 239 184 106)(5 240 185 107)(6 241 186 108)(7 242 187 109)(8 243 188 110)(9 244 189 111)(10 245 190 112)(11 246 191 113)(12 247 192 114)(13 235 193 115)(14 256 37 219)(15 257 38 220)(16 258 39 221)(17 259 27 209)(18 260 28 210)(19 248 29 211)(20 249 30 212)(21 250 31 213)(22 251 32 214)(23 252 33 215)(24 253 34 216)(25 254 35 217)(26 255 36 218)(40 165 132 55)(41 166 133 56)(42 167 134 57)(43 168 135 58)(44 169 136 59)(45 157 137 60)(46 158 138 61)(47 159 139 62)(48 160 140 63)(49 161 141 64)(50 162 142 65)(51 163 143 53)(52 164 131 54)(66 200 84 234)(67 201 85 222)(68 202 86 223)(69 203 87 224)(70 204 88 225)(71 205 89 226)(72 206 90 227)(73 207 91 228)(74 208 79 229)(75 196 80 230)(76 197 81 231)(77 198 82 232)(78 199 83 233)(92 172 129 144)(93 173 130 145)(94 174 118 146)(95 175 119 147)(96 176 120 148)(97 177 121 149)(98 178 122 150)(99 179 123 151)(100 180 124 152)(101 181 125 153)(102 182 126 154)(103 170 127 155)(104 171 128 156)

G:=sub<Sym(260)| (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)(235,236,237,238,239,240,241,242,243,244,245,246,247)(248,249,250,251,252,253,254,255,256,257,258,259,260), (1,93,63,17,225,194,130,160,27,204)(2,94,64,18,226,195,118,161,28,205)(3,95,65,19,227,183,119,162,29,206)(4,96,53,20,228,184,120,163,30,207)(5,97,54,21,229,185,121,164,31,208)(6,98,55,22,230,186,122,165,32,196)(7,99,56,23,231,187,123,166,33,197)(8,100,57,24,232,188,124,167,34,198)(9,101,58,25,233,189,125,168,35,199)(10,102,59,26,234,190,126,169,36,200)(11,103,60,14,222,191,127,157,37,201)(12,104,61,15,223,192,128,158,38,202)(13,92,62,16,224,193,129,159,39,203)(40,178,241,80,214,132,150,108,75,251)(41,179,242,81,215,133,151,109,76,252)(42,180,243,82,216,134,152,110,77,253)(43,181,244,83,217,135,153,111,78,254)(44,182,245,84,218,136,154,112,66,255)(45,170,246,85,219,137,155,113,67,256)(46,171,247,86,220,138,156,114,68,257)(47,172,235,87,221,139,144,115,69,258)(48,173,236,88,209,140,145,116,70,259)(49,174,237,89,210,141,146,117,71,260)(50,175,238,90,211,142,147,105,72,248)(51,176,239,91,212,143,148,106,73,249)(52,177,240,79,213,131,149,107,74,250), (1,236,194,116)(2,237,195,117)(3,238,183,105)(4,239,184,106)(5,240,185,107)(6,241,186,108)(7,242,187,109)(8,243,188,110)(9,244,189,111)(10,245,190,112)(11,246,191,113)(12,247,192,114)(13,235,193,115)(14,256,37,219)(15,257,38,220)(16,258,39,221)(17,259,27,209)(18,260,28,210)(19,248,29,211)(20,249,30,212)(21,250,31,213)(22,251,32,214)(23,252,33,215)(24,253,34,216)(25,254,35,217)(26,255,36,218)(40,165,132,55)(41,166,133,56)(42,167,134,57)(43,168,135,58)(44,169,136,59)(45,157,137,60)(46,158,138,61)(47,159,139,62)(48,160,140,63)(49,161,141,64)(50,162,142,65)(51,163,143,53)(52,164,131,54)(66,200,84,234)(67,201,85,222)(68,202,86,223)(69,203,87,224)(70,204,88,225)(71,205,89,226)(72,206,90,227)(73,207,91,228)(74,208,79,229)(75,196,80,230)(76,197,81,231)(77,198,82,232)(78,199,83,233)(92,172,129,144)(93,173,130,145)(94,174,118,146)(95,175,119,147)(96,176,120,148)(97,177,121,149)(98,178,122,150)(99,179,123,151)(100,180,124,152)(101,181,125,153)(102,182,126,154)(103,170,127,155)(104,171,128,156)>;

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)(235,236,237,238,239,240,241,242,243,244,245,246,247)(248,249,250,251,252,253,254,255,256,257,258,259,260), (1,93,63,17,225,194,130,160,27,204)(2,94,64,18,226,195,118,161,28,205)(3,95,65,19,227,183,119,162,29,206)(4,96,53,20,228,184,120,163,30,207)(5,97,54,21,229,185,121,164,31,208)(6,98,55,22,230,186,122,165,32,196)(7,99,56,23,231,187,123,166,33,197)(8,100,57,24,232,188,124,167,34,198)(9,101,58,25,233,189,125,168,35,199)(10,102,59,26,234,190,126,169,36,200)(11,103,60,14,222,191,127,157,37,201)(12,104,61,15,223,192,128,158,38,202)(13,92,62,16,224,193,129,159,39,203)(40,178,241,80,214,132,150,108,75,251)(41,179,242,81,215,133,151,109,76,252)(42,180,243,82,216,134,152,110,77,253)(43,181,244,83,217,135,153,111,78,254)(44,182,245,84,218,136,154,112,66,255)(45,170,246,85,219,137,155,113,67,256)(46,171,247,86,220,138,156,114,68,257)(47,172,235,87,221,139,144,115,69,258)(48,173,236,88,209,140,145,116,70,259)(49,174,237,89,210,141,146,117,71,260)(50,175,238,90,211,142,147,105,72,248)(51,176,239,91,212,143,148,106,73,249)(52,177,240,79,213,131,149,107,74,250), (1,236,194,116)(2,237,195,117)(3,238,183,105)(4,239,184,106)(5,240,185,107)(6,241,186,108)(7,242,187,109)(8,243,188,110)(9,244,189,111)(10,245,190,112)(11,246,191,113)(12,247,192,114)(13,235,193,115)(14,256,37,219)(15,257,38,220)(16,258,39,221)(17,259,27,209)(18,260,28,210)(19,248,29,211)(20,249,30,212)(21,250,31,213)(22,251,32,214)(23,252,33,215)(24,253,34,216)(25,254,35,217)(26,255,36,218)(40,165,132,55)(41,166,133,56)(42,167,134,57)(43,168,135,58)(44,169,136,59)(45,157,137,60)(46,158,138,61)(47,159,139,62)(48,160,140,63)(49,161,141,64)(50,162,142,65)(51,163,143,53)(52,164,131,54)(66,200,84,234)(67,201,85,222)(68,202,86,223)(69,203,87,224)(70,204,88,225)(71,205,89,226)(72,206,90,227)(73,207,91,228)(74,208,79,229)(75,196,80,230)(76,197,81,231)(77,198,82,232)(78,199,83,233)(92,172,129,144)(93,173,130,145)(94,174,118,146)(95,175,119,147)(96,176,120,148)(97,177,121,149)(98,178,122,150)(99,179,123,151)(100,180,124,152)(101,181,125,153)(102,182,126,154)(103,170,127,155)(104,171,128,156) );

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),(235,236,237,238,239,240,241,242,243,244,245,246,247),(248,249,250,251,252,253,254,255,256,257,258,259,260)], [(1,93,63,17,225,194,130,160,27,204),(2,94,64,18,226,195,118,161,28,205),(3,95,65,19,227,183,119,162,29,206),(4,96,53,20,228,184,120,163,30,207),(5,97,54,21,229,185,121,164,31,208),(6,98,55,22,230,186,122,165,32,196),(7,99,56,23,231,187,123,166,33,197),(8,100,57,24,232,188,124,167,34,198),(9,101,58,25,233,189,125,168,35,199),(10,102,59,26,234,190,126,169,36,200),(11,103,60,14,222,191,127,157,37,201),(12,104,61,15,223,192,128,158,38,202),(13,92,62,16,224,193,129,159,39,203),(40,178,241,80,214,132,150,108,75,251),(41,179,242,81,215,133,151,109,76,252),(42,180,243,82,216,134,152,110,77,253),(43,181,244,83,217,135,153,111,78,254),(44,182,245,84,218,136,154,112,66,255),(45,170,246,85,219,137,155,113,67,256),(46,171,247,86,220,138,156,114,68,257),(47,172,235,87,221,139,144,115,69,258),(48,173,236,88,209,140,145,116,70,259),(49,174,237,89,210,141,146,117,71,260),(50,175,238,90,211,142,147,105,72,248),(51,176,239,91,212,143,148,106,73,249),(52,177,240,79,213,131,149,107,74,250)], [(1,236,194,116),(2,237,195,117),(3,238,183,105),(4,239,184,106),(5,240,185,107),(6,241,186,108),(7,242,187,109),(8,243,188,110),(9,244,189,111),(10,245,190,112),(11,246,191,113),(12,247,192,114),(13,235,193,115),(14,256,37,219),(15,257,38,220),(16,258,39,221),(17,259,27,209),(18,260,28,210),(19,248,29,211),(20,249,30,212),(21,250,31,213),(22,251,32,214),(23,252,33,215),(24,253,34,216),(25,254,35,217),(26,255,36,218),(40,165,132,55),(41,166,133,56),(42,167,134,57),(43,168,135,58),(44,169,136,59),(45,157,137,60),(46,158,138,61),(47,159,139,62),(48,160,140,63),(49,161,141,64),(50,162,142,65),(51,163,143,53),(52,164,131,54),(66,200,84,234),(67,201,85,222),(68,202,86,223),(69,203,87,224),(70,204,88,225),(71,205,89,226),(72,206,90,227),(73,207,91,228),(74,208,79,229),(75,196,80,230),(76,197,81,231),(77,198,82,232),(78,199,83,233),(92,172,129,144),(93,173,130,145),(94,174,118,146),(95,175,119,147),(96,176,120,148),(97,177,121,149),(98,178,122,150),(99,179,123,151),(100,180,124,152),(101,181,125,153),(102,182,126,154),(103,170,127,155),(104,171,128,156)])

104 conjugacy classes

class 1  2 4A4B5A5B10A10B13A···13L26A···26L52A···52X65A···65X130A···130X
order124455101013···1326···2652···5265···65130···130
size115522221···11···15···52···22···2

104 irreducible representations

dim1111112222
type+++-
imageC1C2C4C13C26C52D5Dic5D5×C13C13×Dic5
kernelC13×Dic5C130C65Dic5C10C5C26C13C2C1
# reps112121224222424

Matrix representation of C13×Dic5 in GL3(𝔽521) generated by

100
04230
00423
,
52000
05201
098422
,
28600
043839
014483
G:=sub<GL(3,GF(521))| [1,0,0,0,423,0,0,0,423],[520,0,0,0,520,98,0,1,422],[286,0,0,0,438,144,0,39,83] >;

C13×Dic5 in GAP, Magma, Sage, TeX

C_{13}\times {\rm Dic}_5
% in TeX

G:=Group("C13xDic5");
// GroupNames label

G:=SmallGroup(260,1);
// by ID

G=gap.SmallGroup(260,1);
# by ID

G:=PCGroup([4,-2,-13,-2,-5,104,3331]);
// Polycyclic

G:=Group<a,b,c|a^13=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C13×Dic5 in TeX

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