metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.20D60, C40.69D6, C8.16D30, D120⋊15C2, C20.38D12, C60.168D4, C12.38D20, C24.69D10, C22.1D60, Dic60⋊15C2, C60.246C23, C120.81C22, D60.36C22, Dic30.37C22, (C2×C40)⋊6S3, (C2×C8)⋊4D15, (C2×C24)⋊6D5, C5⋊4(C4○D24), (C2×C120)⋊10C2, C15⋊25(C4○D8), C6.39(C2×D20), C2.13(C2×D60), (C2×C6).20D20, (C2×C4).82D30, C8⋊D15⋊15C2, C3⋊4(D40⋊7C2), C30.267(C2×D4), (C2×C30).105D4, C10.40(C2×D12), (C2×C20).400D6, (C2×C10).20D12, D60⋊11C2⋊1C2, (C2×C12).405D10, C4.27(C22×D15), (C2×C60).486C22, C20.217(C22×S3), C12.219(C22×D5), SmallGroup(480,869)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.20D60
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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5 [×2], C10, C10, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, D15 [×2], C30, C30, C4○D8, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C24⋊C2 [×2], D24, Dic12, C2×C24, C4○D12 [×2], Dic15 [×2], C60 [×2], D30 [×2], C2×C30, C40⋊C2 [×2], D40, Dic20, C2×C40, C4○D20 [×2], C4○D24, C120 [×2], Dic30 [×2], C4×D15 [×2], D60 [×2], C15⋊7D4 [×2], C2×C60, D40⋊7C2, C8⋊D15 [×2], D120, Dic60, C2×C120, D60⋊11C2 [×2], C4.20D60
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, D15, C4○D8, D20 [×2], C22×D5, C2×D12, D30 [×3], C2×D20, C4○D24, D60 [×2], C22×D15, D40⋊7C2, C2×D60, C4.20D60
►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)
(1 208 61 148)(2 147 62 207)(3 206 63 146)(4 145 64 205)(5 204 65 144)(6 143 66 203)(7 202 67 142)(8 141 68 201)(9 200 69 140)(10 139 70 199)(11 198 71 138)(12 137 72 197)(13 196 73 136)(14 135 74 195)(15 194 75 134)(16 133 76 193)(17 192 77 132)(18 131 78 191)(19 190 79 130)(20 129 80 189)(21 188 81 128)(22 127 82 187)(23 186 83 126)(24 125 84 185)(25 184 85 124)(26 123 86 183)(27 182 87 122)(28 121 88 181)(29 180 89 240)(30 239 90 179)(31 178 91 238)(32 237 92 177)(33 176 93 236)(34 235 94 175)(35 174 95 234)(36 233 96 173)(37 172 97 232)(38 231 98 171)(39 170 99 230)(40 229 100 169)(41 168 101 228)(42 227 102 167)(43 166 103 226)(44 225 104 165)(45 164 105 224)(46 223 106 163)(47 162 107 222)(48 221 108 161)(49 160 109 220)(50 219 110 159)(51 158 111 218)(52 217 112 157)(53 156 113 216)(54 215 114 155)(55 154 115 214)(56 213 116 153)(57 152 117 212)(58 211 118 151)(59 150 119 210)(60 209 120 149)
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), (1,208,61,148)(2,147,62,207)(3,206,63,146)(4,145,64,205)(5,204,65,144)(6,143,66,203)(7,202,67,142)(8,141,68,201)(9,200,69,140)(10,139,70,199)(11,198,71,138)(12,137,72,197)(13,196,73,136)(14,135,74,195)(15,194,75,134)(16,133,76,193)(17,192,77,132)(18,131,78,191)(19,190,79,130)(20,129,80,189)(21,188,81,128)(22,127,82,187)(23,186,83,126)(24,125,84,185)(25,184,85,124)(26,123,86,183)(27,182,87,122)(28,121,88,181)(29,180,89,240)(30,239,90,179)(31,178,91,238)(32,237,92,177)(33,176,93,236)(34,235,94,175)(35,174,95,234)(36,233,96,173)(37,172,97,232)(38,231,98,171)(39,170,99,230)(40,229,100,169)(41,168,101,228)(42,227,102,167)(43,166,103,226)(44,225,104,165)(45,164,105,224)(46,223,106,163)(47,162,107,222)(48,221,108,161)(49,160,109,220)(50,219,110,159)(51,158,111,218)(52,217,112,157)(53,156,113,216)(54,215,114,155)(55,154,115,214)(56,213,116,153)(57,152,117,212)(58,211,118,151)(59,150,119,210)(60,209,120,149)>;
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), (1,208,61,148)(2,147,62,207)(3,206,63,146)(4,145,64,205)(5,204,65,144)(6,143,66,203)(7,202,67,142)(8,141,68,201)(9,200,69,140)(10,139,70,199)(11,198,71,138)(12,137,72,197)(13,196,73,136)(14,135,74,195)(15,194,75,134)(16,133,76,193)(17,192,77,132)(18,131,78,191)(19,190,79,130)(20,129,80,189)(21,188,81,128)(22,127,82,187)(23,186,83,126)(24,125,84,185)(25,184,85,124)(26,123,86,183)(27,182,87,122)(28,121,88,181)(29,180,89,240)(30,239,90,179)(31,178,91,238)(32,237,92,177)(33,176,93,236)(34,235,94,175)(35,174,95,234)(36,233,96,173)(37,172,97,232)(38,231,98,171)(39,170,99,230)(40,229,100,169)(41,168,101,228)(42,227,102,167)(43,166,103,226)(44,225,104,165)(45,164,105,224)(46,223,106,163)(47,162,107,222)(48,221,108,161)(49,160,109,220)(50,219,110,159)(51,158,111,218)(52,217,112,157)(53,156,113,216)(54,215,114,155)(55,154,115,214)(56,213,116,153)(57,152,117,212)(58,211,118,151)(59,150,119,210)(60,209,120,149) );
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)], [(1,208,61,148),(2,147,62,207),(3,206,63,146),(4,145,64,205),(5,204,65,144),(6,143,66,203),(7,202,67,142),(8,141,68,201),(9,200,69,140),(10,139,70,199),(11,198,71,138),(12,137,72,197),(13,196,73,136),(14,135,74,195),(15,194,75,134),(16,133,76,193),(17,192,77,132),(18,131,78,191),(19,190,79,130),(20,129,80,189),(21,188,81,128),(22,127,82,187),(23,186,83,126),(24,125,84,185),(25,184,85,124),(26,123,86,183),(27,182,87,122),(28,121,88,181),(29,180,89,240),(30,239,90,179),(31,178,91,238),(32,237,92,177),(33,176,93,236),(34,235,94,175),(35,174,95,234),(36,233,96,173),(37,172,97,232),(38,231,98,171),(39,170,99,230),(40,229,100,169),(41,168,101,228),(42,227,102,167),(43,166,103,226),(44,225,104,165),(45,164,105,224),(46,223,106,163),(47,162,107,222),(48,221,108,161),(49,160,109,220),(50,219,110,159),(51,158,111,218),(52,217,112,157),(53,156,113,216),(54,215,114,155),(55,154,115,214),(56,213,116,153),(57,152,117,212),(58,211,118,151),(59,150,119,210),(60,209,120,149)])
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 | C4.20D60 |
| kernel | C4.20D60 | C8⋊D15 | 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 C4.20D60 ►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] >;
C4.20D60 in GAP, Magma, Sage, TeX
C_4._{20}D_{60} % in TeX
G:=Group("C4.20D60"); // 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