metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.69D6, C4.20D60, C8.16D30, D120⋊15C2, C60.168D4, C20.38D12, C12.38D20, C24.69D10, C22.1D60, Dic60⋊15C2, C120.81C22, C60.246C23, D60.36C22, Dic30.37C22, (C2×C40)⋊6S3, (C2×C24)⋊6D5, (C2×C8)⋊4D15, C5⋊4(C4○D24), (C2×C120)⋊10C2, C15⋊25(C4○D8), (C2×C4).82D30, C2.13(C2×D60), (C2×C6).20D20, C6.39(C2×D20), C24⋊D5⋊15C2, C3⋊4(D40⋊7C2), (C2×C20).400D6, (C2×C10).20D12, C10.40(C2×D12), (C2×C30).105D4, C30.267(C2×D4), D60⋊11C2⋊1C2, (C2×C12).405D10, C4.27(C22×D15), C20.217(C22×S3), (C2×C60).486C22, C12.219(C22×D5), SmallGroup(480,869)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.69D6
G = < a,b,c | a4=1, b60=c2=a2, ab=ba, ac=ca, cbc-1=b59 >
Subgroups: 884 in 124 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, C2×C10, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, D15, C30, C30, C4○D8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C24⋊C2, D24, Dic12, C2×C24, C4○D12, Dic15, C60, D30, C2×C30, C40⋊C2, D40, Dic20, C2×C40, C4○D20, C4○D24, C120, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D40⋊7C2, C24⋊D5, D120, Dic60, C2×C120, D60⋊11C2, C40.69D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, D15, C4○D8, D20, C22×D5, C2×D12, D30, C2×D20, C4○D24, D60, C22×D15, D40⋊7C2, C2×D60, C40.69D6
►On 240 points
Generators in S240
(1 31 61 91)(2 32 62 92)(3 33 63 93)(4 34 64 94)(5 35 65 95)(6 36 66 96)(7 37 67 97)(8 38 68 98)(9 39 69 99)(10 40 70 100)(11 41 71 101)(12 42 72 102)(13 43 73 103)(14 44 74 104)(15 45 75 105)(16 46 76 106)(17 47 77 107)(18 48 78 108)(19 49 79 109)(20 50 80 110)(21 51 81 111)(22 52 82 112)(23 53 83 113)(24 54 84 114)(25 55 85 115)(26 56 86 116)(27 57 87 117)(28 58 88 118)(29 59 89 119)(30 60 90 120)(121 211 181 151)(122 212 182 152)(123 213 183 153)(124 214 184 154)(125 215 185 155)(126 216 186 156)(127 217 187 157)(128 218 188 158)(129 219 189 159)(130 220 190 160)(131 221 191 161)(132 222 192 162)(133 223 193 163)(134 224 194 164)(135 225 195 165)(136 226 196 166)(137 227 197 167)(138 228 198 168)(139 229 199 169)(140 230 200 170)(141 231 201 171)(142 232 202 172)(143 233 203 173)(144 234 204 174)(145 235 205 175)(146 236 206 176)(147 237 207 177)(148 238 208 178)(149 239 209 179)(150 240 210 180)
(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)
(1 150 61 210)(2 209 62 149)(3 148 63 208)(4 207 64 147)(5 146 65 206)(6 205 66 145)(7 144 67 204)(8 203 68 143)(9 142 69 202)(10 201 70 141)(11 140 71 200)(12 199 72 139)(13 138 73 198)(14 197 74 137)(15 136 75 196)(16 195 76 135)(17 134 77 194)(18 193 78 133)(19 132 79 192)(20 191 80 131)(21 130 81 190)(22 189 82 129)(23 128 83 188)(24 187 84 127)(25 126 85 186)(26 185 86 125)(27 124 87 184)(28 183 88 123)(29 122 89 182)(30 181 90 121)(31 240 91 180)(32 179 92 239)(33 238 93 178)(34 177 94 237)(35 236 95 176)(36 175 96 235)(37 234 97 174)(38 173 98 233)(39 232 99 172)(40 171 100 231)(41 230 101 170)(42 169 102 229)(43 228 103 168)(44 167 104 227)(45 226 105 166)(46 165 106 225)(47 224 107 164)(48 163 108 223)(49 222 109 162)(50 161 110 221)(51 220 111 160)(52 159 112 219)(53 218 113 158)(54 157 114 217)(55 216 115 156)(56 155 116 215)(57 214 117 154)(58 153 118 213)(59 212 119 152)(60 151 120 211)
G:=sub<Sym(240)| (1,31,61,91)(2,32,62,92)(3,33,63,93)(4,34,64,94)(5,35,65,95)(6,36,66,96)(7,37,67,97)(8,38,68,98)(9,39,69,99)(10,40,70,100)(11,41,71,101)(12,42,72,102)(13,43,73,103)(14,44,74,104)(15,45,75,105)(16,46,76,106)(17,47,77,107)(18,48,78,108)(19,49,79,109)(20,50,80,110)(21,51,81,111)(22,52,82,112)(23,53,83,113)(24,54,84,114)(25,55,85,115)(26,56,86,116)(27,57,87,117)(28,58,88,118)(29,59,89,119)(30,60,90,120)(121,211,181,151)(122,212,182,152)(123,213,183,153)(124,214,184,154)(125,215,185,155)(126,216,186,156)(127,217,187,157)(128,218,188,158)(129,219,189,159)(130,220,190,160)(131,221,191,161)(132,222,192,162)(133,223,193,163)(134,224,194,164)(135,225,195,165)(136,226,196,166)(137,227,197,167)(138,228,198,168)(139,229,199,169)(140,230,200,170)(141,231,201,171)(142,232,202,172)(143,233,203,173)(144,234,204,174)(145,235,205,175)(146,236,206,176)(147,237,207,177)(148,238,208,178)(149,239,209,179)(150,240,210,180), (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), (1,150,61,210)(2,209,62,149)(3,148,63,208)(4,207,64,147)(5,146,65,206)(6,205,66,145)(7,144,67,204)(8,203,68,143)(9,142,69,202)(10,201,70,141)(11,140,71,200)(12,199,72,139)(13,138,73,198)(14,197,74,137)(15,136,75,196)(16,195,76,135)(17,134,77,194)(18,193,78,133)(19,132,79,192)(20,191,80,131)(21,130,81,190)(22,189,82,129)(23,128,83,188)(24,187,84,127)(25,126,85,186)(26,185,86,125)(27,124,87,184)(28,183,88,123)(29,122,89,182)(30,181,90,121)(31,240,91,180)(32,179,92,239)(33,238,93,178)(34,177,94,237)(35,236,95,176)(36,175,96,235)(37,234,97,174)(38,173,98,233)(39,232,99,172)(40,171,100,231)(41,230,101,170)(42,169,102,229)(43,228,103,168)(44,167,104,227)(45,226,105,166)(46,165,106,225)(47,224,107,164)(48,163,108,223)(49,222,109,162)(50,161,110,221)(51,220,111,160)(52,159,112,219)(53,218,113,158)(54,157,114,217)(55,216,115,156)(56,155,116,215)(57,214,117,154)(58,153,118,213)(59,212,119,152)(60,151,120,211)>;
G:=Group( (1,31,61,91)(2,32,62,92)(3,33,63,93)(4,34,64,94)(5,35,65,95)(6,36,66,96)(7,37,67,97)(8,38,68,98)(9,39,69,99)(10,40,70,100)(11,41,71,101)(12,42,72,102)(13,43,73,103)(14,44,74,104)(15,45,75,105)(16,46,76,106)(17,47,77,107)(18,48,78,108)(19,49,79,109)(20,50,80,110)(21,51,81,111)(22,52,82,112)(23,53,83,113)(24,54,84,114)(25,55,85,115)(26,56,86,116)(27,57,87,117)(28,58,88,118)(29,59,89,119)(30,60,90,120)(121,211,181,151)(122,212,182,152)(123,213,183,153)(124,214,184,154)(125,215,185,155)(126,216,186,156)(127,217,187,157)(128,218,188,158)(129,219,189,159)(130,220,190,160)(131,221,191,161)(132,222,192,162)(133,223,193,163)(134,224,194,164)(135,225,195,165)(136,226,196,166)(137,227,197,167)(138,228,198,168)(139,229,199,169)(140,230,200,170)(141,231,201,171)(142,232,202,172)(143,233,203,173)(144,234,204,174)(145,235,205,175)(146,236,206,176)(147,237,207,177)(148,238,208,178)(149,239,209,179)(150,240,210,180), (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), (1,150,61,210)(2,209,62,149)(3,148,63,208)(4,207,64,147)(5,146,65,206)(6,205,66,145)(7,144,67,204)(8,203,68,143)(9,142,69,202)(10,201,70,141)(11,140,71,200)(12,199,72,139)(13,138,73,198)(14,197,74,137)(15,136,75,196)(16,195,76,135)(17,134,77,194)(18,193,78,133)(19,132,79,192)(20,191,80,131)(21,130,81,190)(22,189,82,129)(23,128,83,188)(24,187,84,127)(25,126,85,186)(26,185,86,125)(27,124,87,184)(28,183,88,123)(29,122,89,182)(30,181,90,121)(31,240,91,180)(32,179,92,239)(33,238,93,178)(34,177,94,237)(35,236,95,176)(36,175,96,235)(37,234,97,174)(38,173,98,233)(39,232,99,172)(40,171,100,231)(41,230,101,170)(42,169,102,229)(43,228,103,168)(44,167,104,227)(45,226,105,166)(46,165,106,225)(47,224,107,164)(48,163,108,223)(49,222,109,162)(50,161,110,221)(51,220,111,160)(52,159,112,219)(53,218,113,158)(54,157,114,217)(55,216,115,156)(56,155,116,215)(57,214,117,154)(58,153,118,213)(59,212,119,152)(60,151,120,211) );
G=PermutationGroup([[(1,31,61,91),(2,32,62,92),(3,33,63,93),(4,34,64,94),(5,35,65,95),(6,36,66,96),(7,37,67,97),(8,38,68,98),(9,39,69,99),(10,40,70,100),(11,41,71,101),(12,42,72,102),(13,43,73,103),(14,44,74,104),(15,45,75,105),(16,46,76,106),(17,47,77,107),(18,48,78,108),(19,49,79,109),(20,50,80,110),(21,51,81,111),(22,52,82,112),(23,53,83,113),(24,54,84,114),(25,55,85,115),(26,56,86,116),(27,57,87,117),(28,58,88,118),(29,59,89,119),(30,60,90,120),(121,211,181,151),(122,212,182,152),(123,213,183,153),(124,214,184,154),(125,215,185,155),(126,216,186,156),(127,217,187,157),(128,218,188,158),(129,219,189,159),(130,220,190,160),(131,221,191,161),(132,222,192,162),(133,223,193,163),(134,224,194,164),(135,225,195,165),(136,226,196,166),(137,227,197,167),(138,228,198,168),(139,229,199,169),(140,230,200,170),(141,231,201,171),(142,232,202,172),(143,233,203,173),(144,234,204,174),(145,235,205,175),(146,236,206,176),(147,237,207,177),(148,238,208,178),(149,239,209,179),(150,240,210,180)], [(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)], [(1,150,61,210),(2,209,62,149),(3,148,63,208),(4,207,64,147),(5,146,65,206),(6,205,66,145),(7,144,67,204),(8,203,68,143),(9,142,69,202),(10,201,70,141),(11,140,71,200),(12,199,72,139),(13,138,73,198),(14,197,74,137),(15,136,75,196),(16,195,76,135),(17,134,77,194),(18,193,78,133),(19,132,79,192),(20,191,80,131),(21,130,81,190),(22,189,82,129),(23,128,83,188),(24,187,84,127),(25,126,85,186),(26,185,86,125),(27,124,87,184),(28,183,88,123),(29,122,89,182),(30,181,90,121),(31,240,91,180),(32,179,92,239),(33,238,93,178),(34,177,94,237),(35,236,95,176),(36,175,96,235),(37,234,97,174),(38,173,98,233),(39,232,99,172),(40,171,100,231),(41,230,101,170),(42,169,102,229),(43,228,103,168),(44,167,104,227),(45,226,105,166),(46,165,106,225),(47,224,107,164),(48,163,108,223),(49,222,109,162),(50,161,110,221),(51,220,111,160),(52,159,112,219),(53,218,113,158),(54,157,114,217),(55,216,115,156),(56,155,116,215),(57,214,117,154),(58,153,118,213),(59,212,119,152),(60,151,120,211)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 60 | 60 | 2 | 1 | 1 | 2 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | D12 | D15 | C4○D8 | D20 | D20 | D30 | D30 | C4○D24 | D60 | D60 | D40⋊7C2 | C40.69D6 |
| kernel | C40.69D6 | C24⋊D5 | D120 | Dic60 | C2×C120 | D60⋊11C2 | C2×C40 | C60 | C2×C30 | C2×C24 | C40 | C2×C20 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C15 | C12 | C2×C6 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C40.69D6 ►in GL2(𝔽241) generated by
| 64 | 0 |
| 0 | 64 |
| 133 | 0 |
| 0 | 212 |
| 0 | 212 |
| 133 | 0 |
G:=sub<GL(2,GF(241))| [64,0,0,64],[133,0,0,212],[0,133,212,0] >;
C40.69D6 in GAP, Magma, Sage, TeX
C_{40}._{69}D_6 % in TeX
G:=Group("C40.69D6"); // GroupNames label
G:=SmallGroup(480,869);
// by ID
G=gap.SmallGroup(480,869);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,58,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^60=c^2=a^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^59>;
// generators/relations