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G = C5×Dic13order 260 = 22·5·13

Direct product of C5 and Dic13

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

Aliases: C5×Dic13, C658C4, C132C20, C26.C10, C130.2C2, C10.2D13, C2.(C5×D13), SmallGroup(260,2)

Series: Derived Chief Lower central Upper central

C1C13 — C5×Dic13
C1C13C26C130 — C5×Dic13
C13 — C5×Dic13
C1C10

Generators and relations for C5×Dic13
 G = < a,b,c | a5=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >

13C4
13C20

Smallest permutation representation of C5×Dic13
Regular action on 260 points
Generators in S260
(1 109 103 68 39)(2 110 104 69 40)(3 111 79 70 41)(4 112 80 71 42)(5 113 81 72 43)(6 114 82 73 44)(7 115 83 74 45)(8 116 84 75 46)(9 117 85 76 47)(10 118 86 77 48)(11 119 87 78 49)(12 120 88 53 50)(13 121 89 54 51)(14 122 90 55 52)(15 123 91 56 27)(16 124 92 57 28)(17 125 93 58 29)(18 126 94 59 30)(19 127 95 60 31)(20 128 96 61 32)(21 129 97 62 33)(22 130 98 63 34)(23 105 99 64 35)(24 106 100 65 36)(25 107 101 66 37)(26 108 102 67 38)(131 235 222 196 157)(132 236 223 197 158)(133 237 224 198 159)(134 238 225 199 160)(135 239 226 200 161)(136 240 227 201 162)(137 241 228 202 163)(138 242 229 203 164)(139 243 230 204 165)(140 244 231 205 166)(141 245 232 206 167)(142 246 233 207 168)(143 247 234 208 169)(144 248 209 183 170)(145 249 210 184 171)(146 250 211 185 172)(147 251 212 186 173)(148 252 213 187 174)(149 253 214 188 175)(150 254 215 189 176)(151 255 216 190 177)(152 256 217 191 178)(153 257 218 192 179)(154 258 219 193 180)(155 259 220 194 181)(156 260 221 195 182)
(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)
(1 131 14 144)(2 156 15 143)(3 155 16 142)(4 154 17 141)(5 153 18 140)(6 152 19 139)(7 151 20 138)(8 150 21 137)(9 149 22 136)(10 148 23 135)(11 147 24 134)(12 146 25 133)(13 145 26 132)(27 169 40 182)(28 168 41 181)(29 167 42 180)(30 166 43 179)(31 165 44 178)(32 164 45 177)(33 163 46 176)(34 162 47 175)(35 161 48 174)(36 160 49 173)(37 159 50 172)(38 158 51 171)(39 157 52 170)(53 185 66 198)(54 184 67 197)(55 183 68 196)(56 208 69 195)(57 207 70 194)(58 206 71 193)(59 205 72 192)(60 204 73 191)(61 203 74 190)(62 202 75 189)(63 201 76 188)(64 200 77 187)(65 199 78 186)(79 220 92 233)(80 219 93 232)(81 218 94 231)(82 217 95 230)(83 216 96 229)(84 215 97 228)(85 214 98 227)(86 213 99 226)(87 212 100 225)(88 211 101 224)(89 210 102 223)(90 209 103 222)(91 234 104 221)(105 239 118 252)(106 238 119 251)(107 237 120 250)(108 236 121 249)(109 235 122 248)(110 260 123 247)(111 259 124 246)(112 258 125 245)(113 257 126 244)(114 256 127 243)(115 255 128 242)(116 254 129 241)(117 253 130 240)

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

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

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

80 conjugacy classes

class 1  2 4A4B5A5B5C5D10A10B10C10D13A···13F20A···20H26A···26F65A···65X130A···130X
order124455551010101013···1320···2026···2665···65130···130
size111313111111112···213···132···22···22···2

80 irreducible representations

dim1111112222
type+++-
imageC1C2C4C5C10C20D13Dic13C5×D13C5×Dic13
kernelC5×Dic13C130C65Dic13C26C13C10C5C2C1
# reps112448662424

Matrix representation of C5×Dic13 in GL2(𝔽521) generated by

3960
0396
,
0520
1474
,
39289
38482
G:=sub<GL(2,GF(521))| [396,0,0,396],[0,1,520,474],[39,38,289,482] >;

C5×Dic13 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{13}
% in TeX

G:=Group("C5xDic13");
// GroupNames label

G:=SmallGroup(260,2);
// by ID

G=gap.SmallGroup(260,2);
# by ID

G:=PCGroup([4,-2,-5,-2,-13,40,3843]);
// Polycyclic

G:=Group<a,b,c|a^5=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×Dic13 in TeX

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