metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.12D60, C20.12D12, C60.204D4, C12.12D20, M4(2).2D15, (C2×C4).2D30, (C2×C20).70D6, (C2×C12).71D10, C22.5(C4×D15), C10.35(D6⋊C4), C60.7C4.7C2, C15⋊7(C4.10D4), C5⋊3(C12.47D4), C4.22(C15⋊7D4), C3⋊2(C4.12D20), (C2×C60).56C22, (C2×Dic15).1C4, (C2×Dic30).7C2, (C5×M4(2)).4S3, (C3×M4(2)).4D5, C12.101(C5⋊D4), C20.101(C3⋊D4), C30.77(C22⋊C4), (C15×M4(2)).6C2, C6.20(D10⋊C4), C2.10(D30⋊3C4), (C2×C6).6(C4×D5), (C2×C30).66(C2×C4), (C2×C10).29(C4×S3), SmallGroup(480,184)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.D60
G = < a,b,c | a4=1, b60=a2, c2=bab-1=a-1, ac=ca, cbc-1=a-1b59 >
Subgroups: 404 in 76 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, M4(2), M4(2), C2×Q8, Dic5, C20, C2×C10, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C30, C30, C4.10D4, C5⋊2C8, C40, Dic10, C2×Dic5, C2×C20, C4.Dic3, C3×M4(2), C2×Dic6, Dic15, C60, C2×C30, C4.Dic5, C5×M4(2), C2×Dic10, C12.47D4, C15⋊3C8, C120, Dic30, C2×Dic15, C2×C60, C4.12D20, C60.7C4, C15×M4(2), C2×Dic30, C4.D60
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4.10D4, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C12.47D4, C4×D15, D60, C15⋊7D4, C4.12D20, D30⋊3C4, C4.D60
►On 240 points
Generators in S240
(1 91 61 31)(2 32 62 92)(3 93 63 33)(4 34 64 94)(5 95 65 35)(6 36 66 96)(7 97 67 37)(8 38 68 98)(9 99 69 39)(10 40 70 100)(11 101 71 41)(12 42 72 102)(13 103 73 43)(14 44 74 104)(15 105 75 45)(16 46 76 106)(17 107 77 47)(18 48 78 108)(19 109 79 49)(20 50 80 110)(21 111 81 51)(22 52 82 112)(23 113 83 53)(24 54 84 114)(25 115 85 55)(26 56 86 116)(27 117 87 57)(28 58 88 118)(29 119 89 59)(30 60 90 120)(121 151 181 211)(122 212 182 152)(123 153 183 213)(124 214 184 154)(125 155 185 215)(126 216 186 156)(127 157 187 217)(128 218 188 158)(129 159 189 219)(130 220 190 160)(131 161 191 221)(132 222 192 162)(133 163 193 223)(134 224 194 164)(135 165 195 225)(136 226 196 166)(137 167 197 227)(138 228 198 168)(139 169 199 229)(140 230 200 170)(141 171 201 231)(142 232 202 172)(143 173 203 233)(144 234 204 174)(145 175 205 235)(146 236 206 176)(147 177 207 237)(148 238 208 178)(149 179 209 239)(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)
(1 161 31 131 61 221 91 191)(2 130 92 160 62 190 32 220)(3 159 33 129 63 219 93 189)(4 128 94 158 64 188 34 218)(5 157 35 127 65 217 95 187)(6 126 96 156 66 186 36 216)(7 155 37 125 67 215 97 185)(8 124 98 154 68 184 38 214)(9 153 39 123 69 213 99 183)(10 122 100 152 70 182 40 212)(11 151 41 121 71 211 101 181)(12 240 102 150 72 180 42 210)(13 149 43 239 73 209 103 179)(14 238 104 148 74 178 44 208)(15 147 45 237 75 207 105 177)(16 236 106 146 76 176 46 206)(17 145 47 235 77 205 107 175)(18 234 108 144 78 174 48 204)(19 143 49 233 79 203 109 173)(20 232 110 142 80 172 50 202)(21 141 51 231 81 201 111 171)(22 230 112 140 82 170 52 200)(23 139 53 229 83 199 113 169)(24 228 114 138 84 168 54 198)(25 137 55 227 85 197 115 167)(26 226 116 136 86 166 56 196)(27 135 57 225 87 195 117 165)(28 224 118 134 88 164 58 194)(29 133 59 223 89 193 119 163)(30 222 120 132 90 162 60 192)
G:=sub<Sym(240)| (1,91,61,31)(2,32,62,92)(3,93,63,33)(4,34,64,94)(5,95,65,35)(6,36,66,96)(7,97,67,37)(8,38,68,98)(9,99,69,39)(10,40,70,100)(11,101,71,41)(12,42,72,102)(13,103,73,43)(14,44,74,104)(15,105,75,45)(16,46,76,106)(17,107,77,47)(18,48,78,108)(19,109,79,49)(20,50,80,110)(21,111,81,51)(22,52,82,112)(23,113,83,53)(24,54,84,114)(25,115,85,55)(26,56,86,116)(27,117,87,57)(28,58,88,118)(29,119,89,59)(30,60,90,120)(121,151,181,211)(122,212,182,152)(123,153,183,213)(124,214,184,154)(125,155,185,215)(126,216,186,156)(127,157,187,217)(128,218,188,158)(129,159,189,219)(130,220,190,160)(131,161,191,221)(132,222,192,162)(133,163,193,223)(134,224,194,164)(135,165,195,225)(136,226,196,166)(137,167,197,227)(138,228,198,168)(139,169,199,229)(140,230,200,170)(141,171,201,231)(142,232,202,172)(143,173,203,233)(144,234,204,174)(145,175,205,235)(146,236,206,176)(147,177,207,237)(148,238,208,178)(149,179,209,239)(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), (1,161,31,131,61,221,91,191)(2,130,92,160,62,190,32,220)(3,159,33,129,63,219,93,189)(4,128,94,158,64,188,34,218)(5,157,35,127,65,217,95,187)(6,126,96,156,66,186,36,216)(7,155,37,125,67,215,97,185)(8,124,98,154,68,184,38,214)(9,153,39,123,69,213,99,183)(10,122,100,152,70,182,40,212)(11,151,41,121,71,211,101,181)(12,240,102,150,72,180,42,210)(13,149,43,239,73,209,103,179)(14,238,104,148,74,178,44,208)(15,147,45,237,75,207,105,177)(16,236,106,146,76,176,46,206)(17,145,47,235,77,205,107,175)(18,234,108,144,78,174,48,204)(19,143,49,233,79,203,109,173)(20,232,110,142,80,172,50,202)(21,141,51,231,81,201,111,171)(22,230,112,140,82,170,52,200)(23,139,53,229,83,199,113,169)(24,228,114,138,84,168,54,198)(25,137,55,227,85,197,115,167)(26,226,116,136,86,166,56,196)(27,135,57,225,87,195,117,165)(28,224,118,134,88,164,58,194)(29,133,59,223,89,193,119,163)(30,222,120,132,90,162,60,192)>;
G:=Group( (1,91,61,31)(2,32,62,92)(3,93,63,33)(4,34,64,94)(5,95,65,35)(6,36,66,96)(7,97,67,37)(8,38,68,98)(9,99,69,39)(10,40,70,100)(11,101,71,41)(12,42,72,102)(13,103,73,43)(14,44,74,104)(15,105,75,45)(16,46,76,106)(17,107,77,47)(18,48,78,108)(19,109,79,49)(20,50,80,110)(21,111,81,51)(22,52,82,112)(23,113,83,53)(24,54,84,114)(25,115,85,55)(26,56,86,116)(27,117,87,57)(28,58,88,118)(29,119,89,59)(30,60,90,120)(121,151,181,211)(122,212,182,152)(123,153,183,213)(124,214,184,154)(125,155,185,215)(126,216,186,156)(127,157,187,217)(128,218,188,158)(129,159,189,219)(130,220,190,160)(131,161,191,221)(132,222,192,162)(133,163,193,223)(134,224,194,164)(135,165,195,225)(136,226,196,166)(137,167,197,227)(138,228,198,168)(139,169,199,229)(140,230,200,170)(141,171,201,231)(142,232,202,172)(143,173,203,233)(144,234,204,174)(145,175,205,235)(146,236,206,176)(147,177,207,237)(148,238,208,178)(149,179,209,239)(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), (1,161,31,131,61,221,91,191)(2,130,92,160,62,190,32,220)(3,159,33,129,63,219,93,189)(4,128,94,158,64,188,34,218)(5,157,35,127,65,217,95,187)(6,126,96,156,66,186,36,216)(7,155,37,125,67,215,97,185)(8,124,98,154,68,184,38,214)(9,153,39,123,69,213,99,183)(10,122,100,152,70,182,40,212)(11,151,41,121,71,211,101,181)(12,240,102,150,72,180,42,210)(13,149,43,239,73,209,103,179)(14,238,104,148,74,178,44,208)(15,147,45,237,75,207,105,177)(16,236,106,146,76,176,46,206)(17,145,47,235,77,205,107,175)(18,234,108,144,78,174,48,204)(19,143,49,233,79,203,109,173)(20,232,110,142,80,172,50,202)(21,141,51,231,81,201,111,171)(22,230,112,140,82,170,52,200)(23,139,53,229,83,199,113,169)(24,228,114,138,84,168,54,198)(25,137,55,227,85,197,115,167)(26,226,116,136,86,166,56,196)(27,135,57,225,87,195,117,165)(28,224,118,134,88,164,58,194)(29,133,59,223,89,193,119,163)(30,222,120,132,90,162,60,192) );
G=PermutationGroup([[(1,91,61,31),(2,32,62,92),(3,93,63,33),(4,34,64,94),(5,95,65,35),(6,36,66,96),(7,97,67,37),(8,38,68,98),(9,99,69,39),(10,40,70,100),(11,101,71,41),(12,42,72,102),(13,103,73,43),(14,44,74,104),(15,105,75,45),(16,46,76,106),(17,107,77,47),(18,48,78,108),(19,109,79,49),(20,50,80,110),(21,111,81,51),(22,52,82,112),(23,113,83,53),(24,54,84,114),(25,115,85,55),(26,56,86,116),(27,117,87,57),(28,58,88,118),(29,119,89,59),(30,60,90,120),(121,151,181,211),(122,212,182,152),(123,153,183,213),(124,214,184,154),(125,155,185,215),(126,216,186,156),(127,157,187,217),(128,218,188,158),(129,159,189,219),(130,220,190,160),(131,161,191,221),(132,222,192,162),(133,163,193,223),(134,224,194,164),(135,165,195,225),(136,226,196,166),(137,167,197,227),(138,228,198,168),(139,169,199,229),(140,230,200,170),(141,171,201,231),(142,232,202,172),(143,173,203,233),(144,234,204,174),(145,175,205,235),(146,236,206,176),(147,177,207,237),(148,238,208,178),(149,179,209,239),(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)], [(1,161,31,131,61,221,91,191),(2,130,92,160,62,190,32,220),(3,159,33,129,63,219,93,189),(4,128,94,158,64,188,34,218),(5,157,35,127,65,217,95,187),(6,126,96,156,66,186,36,216),(7,155,37,125,67,215,97,185),(8,124,98,154,68,184,38,214),(9,153,39,123,69,213,99,183),(10,122,100,152,70,182,40,212),(11,151,41,121,71,211,101,181),(12,240,102,150,72,180,42,210),(13,149,43,239,73,209,103,179),(14,238,104,148,74,178,44,208),(15,147,45,237,75,207,105,177),(16,236,106,146,76,176,46,206),(17,145,47,235,77,205,107,175),(18,234,108,144,78,174,48,204),(19,143,49,233,79,203,109,173),(20,232,110,142,80,172,50,202),(21,141,51,231,81,201,111,171),(22,230,112,140,82,170,52,200),(23,139,53,229,83,199,113,169),(24,228,114,138,84,168,54,198),(25,137,55,227,85,197,115,167),(26,226,116,136,86,166,56,196),(27,135,57,225,87,195,117,165),(28,224,118,134,88,164,58,194),(29,133,59,223,89,193,119,163),(30,222,120,132,90,162,60,192)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 60 | 60 | 2 | 2 | 2 | 4 | 4 | 4 | 60 | 60 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D10 | D12 | C3⋊D4 | C4×S3 | D15 | D20 | C5⋊D4 | C4×D5 | D30 | D60 | C15⋊7D4 | C4×D15 | C4.10D4 | C12.47D4 | C4.12D20 | C4.D60 |
| kernel | C4.D60 | C60.7C4 | C15×M4(2) | C2×Dic30 | C2×Dic15 | C5×M4(2) | C60 | C3×M4(2) | C2×C20 | C2×C12 | C20 | C20 | C2×C10 | M4(2) | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C4.D60 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 115 | 153 | 0 | 0 |
| 0 | 0 | 101 | 126 | 0 | 0 |
| 0 | 0 | 18 | 16 | 99 | 198 |
| 0 | 0 | 172 | 187 | 43 | 142 |
| 107 | 19 | 0 | 0 | 0 | 0 |
| 222 | 107 | 0 | 0 | 0 | 0 |
| 0 | 0 | 89 | 81 | 236 | 236 |
| 0 | 0 | 82 | 72 | 21 | 70 |
| 0 | 0 | 237 | 129 | 101 | 102 |
| 0 | 0 | 151 | 205 | 102 | 220 |
| 207 | 110 | 0 | 0 | 0 | 0 |
| 110 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 196 | 191 | 207 | 115 |
| 0 | 0 | 97 | 38 | 28 | 33 |
| 0 | 0 | 44 | 182 | 230 | 9 |
| 0 | 0 | 24 | 18 | 4 | 18 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,115,101,18,172,0,0,153,126,16,187,0,0,0,0,99,43,0,0,0,0,198,142],[107,222,0,0,0,0,19,107,0,0,0,0,0,0,89,82,237,151,0,0,81,72,129,205,0,0,236,21,101,102,0,0,236,70,102,220],[207,110,0,0,0,0,110,34,0,0,0,0,0,0,196,97,44,24,0,0,191,38,182,18,0,0,207,28,230,4,0,0,115,33,9,18] >;
C4.D60 in GAP, Magma, Sage, TeX
C_4.D_{60} % in TeX
G:=Group("C4.D60"); // GroupNames label
G:=SmallGroup(480,184);
// by ID
G=gap.SmallGroup(480,184);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,36,422,100,346,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^60=a^2,c^2=b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^59>;
// generators/relations