direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic12, C15⋊6Q16, C40.3S3, C24.1C10, C120.4C2, C30.30D4, C20.54D6, C10.15D12, C60.66C22, Dic6.1C10, C8.(C5×S3), C3⋊1(C5×Q16), C6.3(C5×D4), C2.5(C5×D12), C4.10(S3×C10), C12.10(C2×C10), (C5×Dic6).3C2, SmallGroup(240,53)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic12
G = < a,b,c | a5=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic12
►Regular action on 240 points
Generators in S240
(1 86 199 240 146)(2 87 200 217 147)(3 88 201 218 148)(4 89 202 219 149)(5 90 203 220 150)(6 91 204 221 151)(7 92 205 222 152)(8 93 206 223 153)(9 94 207 224 154)(10 95 208 225 155)(11 96 209 226 156)(12 73 210 227 157)(13 74 211 228 158)(14 75 212 229 159)(15 76 213 230 160)(16 77 214 231 161)(17 78 215 232 162)(18 79 216 233 163)(19 80 193 234 164)(20 81 194 235 165)(21 82 195 236 166)(22 83 196 237 167)(23 84 197 238 168)(24 85 198 239 145)(25 179 103 132 51)(26 180 104 133 52)(27 181 105 134 53)(28 182 106 135 54)(29 183 107 136 55)(30 184 108 137 56)(31 185 109 138 57)(32 186 110 139 58)(33 187 111 140 59)(34 188 112 141 60)(35 189 113 142 61)(36 190 114 143 62)(37 191 115 144 63)(38 192 116 121 64)(39 169 117 122 65)(40 170 118 123 66)(41 171 119 124 67)(42 172 120 125 68)(43 173 97 126 69)(44 174 98 127 70)(45 175 99 128 71)(46 176 100 129 72)(47 177 101 130 49)(48 178 102 131 50)
(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)
(1 37 13 25)(2 36 14 48)(3 35 15 47)(4 34 16 46)(5 33 17 45)(6 32 18 44)(7 31 19 43)(8 30 20 42)(9 29 21 41)(10 28 22 40)(11 27 23 39)(12 26 24 38)(49 148 61 160)(50 147 62 159)(51 146 63 158)(52 145 64 157)(53 168 65 156)(54 167 66 155)(55 166 67 154)(56 165 68 153)(57 164 69 152)(58 163 70 151)(59 162 71 150)(60 161 72 149)(73 180 85 192)(74 179 86 191)(75 178 87 190)(76 177 88 189)(77 176 89 188)(78 175 90 187)(79 174 91 186)(80 173 92 185)(81 172 93 184)(82 171 94 183)(83 170 95 182)(84 169 96 181)(97 205 109 193)(98 204 110 216)(99 203 111 215)(100 202 112 214)(101 201 113 213)(102 200 114 212)(103 199 115 211)(104 198 116 210)(105 197 117 209)(106 196 118 208)(107 195 119 207)(108 194 120 206)(121 227 133 239)(122 226 134 238)(123 225 135 237)(124 224 136 236)(125 223 137 235)(126 222 138 234)(127 221 139 233)(128 220 140 232)(129 219 141 231)(130 218 142 230)(131 217 143 229)(132 240 144 228)
G:=sub<Sym(240)| (1,86,199,240,146)(2,87,200,217,147)(3,88,201,218,148)(4,89,202,219,149)(5,90,203,220,150)(6,91,204,221,151)(7,92,205,222,152)(8,93,206,223,153)(9,94,207,224,154)(10,95,208,225,155)(11,96,209,226,156)(12,73,210,227,157)(13,74,211,228,158)(14,75,212,229,159)(15,76,213,230,160)(16,77,214,231,161)(17,78,215,232,162)(18,79,216,233,163)(19,80,193,234,164)(20,81,194,235,165)(21,82,195,236,166)(22,83,196,237,167)(23,84,197,238,168)(24,85,198,239,145)(25,179,103,132,51)(26,180,104,133,52)(27,181,105,134,53)(28,182,106,135,54)(29,183,107,136,55)(30,184,108,137,56)(31,185,109,138,57)(32,186,110,139,58)(33,187,111,140,59)(34,188,112,141,60)(35,189,113,142,61)(36,190,114,143,62)(37,191,115,144,63)(38,192,116,121,64)(39,169,117,122,65)(40,170,118,123,66)(41,171,119,124,67)(42,172,120,125,68)(43,173,97,126,69)(44,174,98,127,70)(45,175,99,128,71)(46,176,100,129,72)(47,177,101,130,49)(48,178,102,131,50), (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), (1,37,13,25)(2,36,14,48)(3,35,15,47)(4,34,16,46)(5,33,17,45)(6,32,18,44)(7,31,19,43)(8,30,20,42)(9,29,21,41)(10,28,22,40)(11,27,23,39)(12,26,24,38)(49,148,61,160)(50,147,62,159)(51,146,63,158)(52,145,64,157)(53,168,65,156)(54,167,66,155)(55,166,67,154)(56,165,68,153)(57,164,69,152)(58,163,70,151)(59,162,71,150)(60,161,72,149)(73,180,85,192)(74,179,86,191)(75,178,87,190)(76,177,88,189)(77,176,89,188)(78,175,90,187)(79,174,91,186)(80,173,92,185)(81,172,93,184)(82,171,94,183)(83,170,95,182)(84,169,96,181)(97,205,109,193)(98,204,110,216)(99,203,111,215)(100,202,112,214)(101,201,113,213)(102,200,114,212)(103,199,115,211)(104,198,116,210)(105,197,117,209)(106,196,118,208)(107,195,119,207)(108,194,120,206)(121,227,133,239)(122,226,134,238)(123,225,135,237)(124,224,136,236)(125,223,137,235)(126,222,138,234)(127,221,139,233)(128,220,140,232)(129,219,141,231)(130,218,142,230)(131,217,143,229)(132,240,144,228)>;
G:=Group( (1,86,199,240,146)(2,87,200,217,147)(3,88,201,218,148)(4,89,202,219,149)(5,90,203,220,150)(6,91,204,221,151)(7,92,205,222,152)(8,93,206,223,153)(9,94,207,224,154)(10,95,208,225,155)(11,96,209,226,156)(12,73,210,227,157)(13,74,211,228,158)(14,75,212,229,159)(15,76,213,230,160)(16,77,214,231,161)(17,78,215,232,162)(18,79,216,233,163)(19,80,193,234,164)(20,81,194,235,165)(21,82,195,236,166)(22,83,196,237,167)(23,84,197,238,168)(24,85,198,239,145)(25,179,103,132,51)(26,180,104,133,52)(27,181,105,134,53)(28,182,106,135,54)(29,183,107,136,55)(30,184,108,137,56)(31,185,109,138,57)(32,186,110,139,58)(33,187,111,140,59)(34,188,112,141,60)(35,189,113,142,61)(36,190,114,143,62)(37,191,115,144,63)(38,192,116,121,64)(39,169,117,122,65)(40,170,118,123,66)(41,171,119,124,67)(42,172,120,125,68)(43,173,97,126,69)(44,174,98,127,70)(45,175,99,128,71)(46,176,100,129,72)(47,177,101,130,49)(48,178,102,131,50), (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), (1,37,13,25)(2,36,14,48)(3,35,15,47)(4,34,16,46)(5,33,17,45)(6,32,18,44)(7,31,19,43)(8,30,20,42)(9,29,21,41)(10,28,22,40)(11,27,23,39)(12,26,24,38)(49,148,61,160)(50,147,62,159)(51,146,63,158)(52,145,64,157)(53,168,65,156)(54,167,66,155)(55,166,67,154)(56,165,68,153)(57,164,69,152)(58,163,70,151)(59,162,71,150)(60,161,72,149)(73,180,85,192)(74,179,86,191)(75,178,87,190)(76,177,88,189)(77,176,89,188)(78,175,90,187)(79,174,91,186)(80,173,92,185)(81,172,93,184)(82,171,94,183)(83,170,95,182)(84,169,96,181)(97,205,109,193)(98,204,110,216)(99,203,111,215)(100,202,112,214)(101,201,113,213)(102,200,114,212)(103,199,115,211)(104,198,116,210)(105,197,117,209)(106,196,118,208)(107,195,119,207)(108,194,120,206)(121,227,133,239)(122,226,134,238)(123,225,135,237)(124,224,136,236)(125,223,137,235)(126,222,138,234)(127,221,139,233)(128,220,140,232)(129,219,141,231)(130,218,142,230)(131,217,143,229)(132,240,144,228) );
G=PermutationGroup([[(1,86,199,240,146),(2,87,200,217,147),(3,88,201,218,148),(4,89,202,219,149),(5,90,203,220,150),(6,91,204,221,151),(7,92,205,222,152),(8,93,206,223,153),(9,94,207,224,154),(10,95,208,225,155),(11,96,209,226,156),(12,73,210,227,157),(13,74,211,228,158),(14,75,212,229,159),(15,76,213,230,160),(16,77,214,231,161),(17,78,215,232,162),(18,79,216,233,163),(19,80,193,234,164),(20,81,194,235,165),(21,82,195,236,166),(22,83,196,237,167),(23,84,197,238,168),(24,85,198,239,145),(25,179,103,132,51),(26,180,104,133,52),(27,181,105,134,53),(28,182,106,135,54),(29,183,107,136,55),(30,184,108,137,56),(31,185,109,138,57),(32,186,110,139,58),(33,187,111,140,59),(34,188,112,141,60),(35,189,113,142,61),(36,190,114,143,62),(37,191,115,144,63),(38,192,116,121,64),(39,169,117,122,65),(40,170,118,123,66),(41,171,119,124,67),(42,172,120,125,68),(43,173,97,126,69),(44,174,98,127,70),(45,175,99,128,71),(46,176,100,129,72),(47,177,101,130,49),(48,178,102,131,50)], [(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)], [(1,37,13,25),(2,36,14,48),(3,35,15,47),(4,34,16,46),(5,33,17,45),(6,32,18,44),(7,31,19,43),(8,30,20,42),(9,29,21,41),(10,28,22,40),(11,27,23,39),(12,26,24,38),(49,148,61,160),(50,147,62,159),(51,146,63,158),(52,145,64,157),(53,168,65,156),(54,167,66,155),(55,166,67,154),(56,165,68,153),(57,164,69,152),(58,163,70,151),(59,162,71,150),(60,161,72,149),(73,180,85,192),(74,179,86,191),(75,178,87,190),(76,177,88,189),(77,176,89,188),(78,175,90,187),(79,174,91,186),(80,173,92,185),(81,172,93,184),(82,171,94,183),(83,170,95,182),(84,169,96,181),(97,205,109,193),(98,204,110,216),(99,203,111,215),(100,202,112,214),(101,201,113,213),(102,200,114,212),(103,199,115,211),(104,198,116,210),(105,197,117,209),(106,196,118,208),(107,195,119,207),(108,194,120,206),(121,227,133,239),(122,226,134,238),(123,225,135,237),(124,224,136,236),(125,223,137,235),(126,222,138,234),(127,221,139,233),(128,220,140,232),(129,219,141,231),(130,218,142,230),(131,217,143,229),(132,240,144,228)]])
C5×Dic12 is a maximal subgroup of
C40.D6 Dic12⋊D5 C15⋊Q32 C5⋊Dic24 C24.2D10 Dic10.D6 D120⋊C2 D40⋊5S3 D30.3D4 C5×S3×Q16
75 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D4 | D6 | Q16 | D12 | C5×S3 | C5×D4 | Dic12 | S3×C10 | C5×Q16 | C5×D12 | C5×Dic12 |
| kernel | C5×Dic12 | C120 | C5×Dic6 | Dic12 | C24 | Dic6 | C40 | C30 | C20 | C15 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C5×Dic12 ►in GL2(𝔽241) generated by
| 205 | 0 |
| 0 | 205 |
| 9 | 114 |
| 127 | 136 |
| 91 | 125 |
| 34 | 150 |
G:=sub<GL(2,GF(241))| [205,0,0,205],[9,127,114,136],[91,34,125,150] >;
C5×Dic12 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{12} % in TeX
G:=Group("C5xDic12"); // GroupNames label
G:=SmallGroup(240,53);
// by ID
G=gap.SmallGroup(240,53);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,240,265,367,1443,69,5765]);
// Polycyclic
G:=Group<a,b,c|a^5=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export