metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.94D4, C20.54D12, C30.15M4(2), C5⋊5(D6⋊C8), D6⋊(C5⋊2C8), (S3×C10)⋊3C8, C10.23(S3×C8), C15⋊6(C22⋊C8), C30.29(C2×C8), (C2×C20).322D6, C10.38(D6⋊C4), (C2×C12).326D10, C12.62(C5⋊D4), C4.26(C5⋊D12), C3⋊1(C20.55D4), C4.27(C15⋊D4), C20.85(C3⋊D4), C2.1(D6⋊Dic5), C10.13(C8⋊S3), C6.2(C4.Dic5), C6.1(C23.D5), C30.39(C22⋊C4), (C2×C60).224C22, (C10×Dic3).14C4, (C2×Dic3).3Dic5, C2.2(D6.Dic5), (C22×S3).2Dic5, C22.10(S3×Dic5), (S3×C2×C4).7D5, (C6×C5⋊2C8)⋊9C2, (C2×C5⋊2C8)⋊9S3, C6.4(C2×C5⋊2C8), C2.4(S3×C5⋊2C8), (S3×C2×C10).8C4, (S3×C2×C20).7C2, (C2×C15⋊3C8)⋊22C2, (C2×C10).69(C4×S3), (C2×C30).78(C2×C4), (C2×C4).227(S3×D5), (C2×C6).11(C2×Dic5), SmallGroup(480,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.94D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a49, cac-1=a29, cbc-1=a15b3 >
Subgroups: 316 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C22⋊C8, C5⋊2C8, C2×C20, C2×C20, C22×C10, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C5⋊2C8, C2×C5⋊2C8, C22×C20, D6⋊C8, C3×C5⋊2C8, C15⋊3C8, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C20.55D4, C6×C5⋊2C8, C2×C15⋊3C8, S3×C2×C20, C60.94D4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D5, D6, C22⋊C4, C2×C8, M4(2), Dic5, D10, C4×S3, D12, C3⋊D4, C22⋊C8, C5⋊2C8, C2×Dic5, C5⋊D4, S3×C8, C8⋊S3, D6⋊C4, S3×D5, C2×C5⋊2C8, C4.Dic5, C23.D5, D6⋊C8, S3×Dic5, C15⋊D4, C5⋊D12, C20.55D4, S3×C5⋊2C8, D6.Dic5, D6⋊Dic5, C60.94D4
►On 240 points
Generators in S240
(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)
(1 107 183 171 31 77 213 141)(2 96 184 160 32 66 214 130)(3 85 185 149 33 115 215 179)(4 74 186 138 34 104 216 168)(5 63 187 127 35 93 217 157)(6 112 188 176 36 82 218 146)(7 101 189 165 37 71 219 135)(8 90 190 154 38 120 220 124)(9 79 191 143 39 109 221 173)(10 68 192 132 40 98 222 162)(11 117 193 121 41 87 223 151)(12 106 194 170 42 76 224 140)(13 95 195 159 43 65 225 129)(14 84 196 148 44 114 226 178)(15 73 197 137 45 103 227 167)(16 62 198 126 46 92 228 156)(17 111 199 175 47 81 229 145)(18 100 200 164 48 70 230 134)(19 89 201 153 49 119 231 123)(20 78 202 142 50 108 232 172)(21 67 203 131 51 97 233 161)(22 116 204 180 52 86 234 150)(23 105 205 169 53 75 235 139)(24 94 206 158 54 64 236 128)(25 83 207 147 55 113 237 177)(26 72 208 136 56 102 238 166)(27 61 209 125 57 91 239 155)(28 110 210 174 58 80 240 144)(29 99 211 163 59 69 181 133)(30 88 212 152 60 118 182 122)
(1 126 46 171 31 156 16 141)(2 155 47 140 32 125 17 170)(3 124 48 169 33 154 18 139)(4 153 49 138 34 123 19 168)(5 122 50 167 35 152 20 137)(6 151 51 136 36 121 21 166)(7 180 52 165 37 150 22 135)(8 149 53 134 38 179 23 164)(9 178 54 163 39 148 24 133)(10 147 55 132 40 177 25 162)(11 176 56 161 41 146 26 131)(12 145 57 130 42 175 27 160)(13 174 58 159 43 144 28 129)(14 143 59 128 44 173 29 158)(15 172 60 157 45 142 30 127)(61 229 106 214 91 199 76 184)(62 198 107 183 92 228 77 213)(63 227 108 212 93 197 78 182)(64 196 109 181 94 226 79 211)(65 225 110 210 95 195 80 240)(66 194 111 239 96 224 81 209)(67 223 112 208 97 193 82 238)(68 192 113 237 98 222 83 207)(69 221 114 206 99 191 84 236)(70 190 115 235 100 220 85 205)(71 219 116 204 101 189 86 234)(72 188 117 233 102 218 87 203)(73 217 118 202 103 187 88 232)(74 186 119 231 104 216 89 201)(75 215 120 200 105 185 90 230)
G:=sub<Sym(240)| (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), (1,107,183,171,31,77,213,141)(2,96,184,160,32,66,214,130)(3,85,185,149,33,115,215,179)(4,74,186,138,34,104,216,168)(5,63,187,127,35,93,217,157)(6,112,188,176,36,82,218,146)(7,101,189,165,37,71,219,135)(8,90,190,154,38,120,220,124)(9,79,191,143,39,109,221,173)(10,68,192,132,40,98,222,162)(11,117,193,121,41,87,223,151)(12,106,194,170,42,76,224,140)(13,95,195,159,43,65,225,129)(14,84,196,148,44,114,226,178)(15,73,197,137,45,103,227,167)(16,62,198,126,46,92,228,156)(17,111,199,175,47,81,229,145)(18,100,200,164,48,70,230,134)(19,89,201,153,49,119,231,123)(20,78,202,142,50,108,232,172)(21,67,203,131,51,97,233,161)(22,116,204,180,52,86,234,150)(23,105,205,169,53,75,235,139)(24,94,206,158,54,64,236,128)(25,83,207,147,55,113,237,177)(26,72,208,136,56,102,238,166)(27,61,209,125,57,91,239,155)(28,110,210,174,58,80,240,144)(29,99,211,163,59,69,181,133)(30,88,212,152,60,118,182,122), (1,126,46,171,31,156,16,141)(2,155,47,140,32,125,17,170)(3,124,48,169,33,154,18,139)(4,153,49,138,34,123,19,168)(5,122,50,167,35,152,20,137)(6,151,51,136,36,121,21,166)(7,180,52,165,37,150,22,135)(8,149,53,134,38,179,23,164)(9,178,54,163,39,148,24,133)(10,147,55,132,40,177,25,162)(11,176,56,161,41,146,26,131)(12,145,57,130,42,175,27,160)(13,174,58,159,43,144,28,129)(14,143,59,128,44,173,29,158)(15,172,60,157,45,142,30,127)(61,229,106,214,91,199,76,184)(62,198,107,183,92,228,77,213)(63,227,108,212,93,197,78,182)(64,196,109,181,94,226,79,211)(65,225,110,210,95,195,80,240)(66,194,111,239,96,224,81,209)(67,223,112,208,97,193,82,238)(68,192,113,237,98,222,83,207)(69,221,114,206,99,191,84,236)(70,190,115,235,100,220,85,205)(71,219,116,204,101,189,86,234)(72,188,117,233,102,218,87,203)(73,217,118,202,103,187,88,232)(74,186,119,231,104,216,89,201)(75,215,120,200,105,185,90,230)>;
G:=Group( (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), (1,107,183,171,31,77,213,141)(2,96,184,160,32,66,214,130)(3,85,185,149,33,115,215,179)(4,74,186,138,34,104,216,168)(5,63,187,127,35,93,217,157)(6,112,188,176,36,82,218,146)(7,101,189,165,37,71,219,135)(8,90,190,154,38,120,220,124)(9,79,191,143,39,109,221,173)(10,68,192,132,40,98,222,162)(11,117,193,121,41,87,223,151)(12,106,194,170,42,76,224,140)(13,95,195,159,43,65,225,129)(14,84,196,148,44,114,226,178)(15,73,197,137,45,103,227,167)(16,62,198,126,46,92,228,156)(17,111,199,175,47,81,229,145)(18,100,200,164,48,70,230,134)(19,89,201,153,49,119,231,123)(20,78,202,142,50,108,232,172)(21,67,203,131,51,97,233,161)(22,116,204,180,52,86,234,150)(23,105,205,169,53,75,235,139)(24,94,206,158,54,64,236,128)(25,83,207,147,55,113,237,177)(26,72,208,136,56,102,238,166)(27,61,209,125,57,91,239,155)(28,110,210,174,58,80,240,144)(29,99,211,163,59,69,181,133)(30,88,212,152,60,118,182,122), (1,126,46,171,31,156,16,141)(2,155,47,140,32,125,17,170)(3,124,48,169,33,154,18,139)(4,153,49,138,34,123,19,168)(5,122,50,167,35,152,20,137)(6,151,51,136,36,121,21,166)(7,180,52,165,37,150,22,135)(8,149,53,134,38,179,23,164)(9,178,54,163,39,148,24,133)(10,147,55,132,40,177,25,162)(11,176,56,161,41,146,26,131)(12,145,57,130,42,175,27,160)(13,174,58,159,43,144,28,129)(14,143,59,128,44,173,29,158)(15,172,60,157,45,142,30,127)(61,229,106,214,91,199,76,184)(62,198,107,183,92,228,77,213)(63,227,108,212,93,197,78,182)(64,196,109,181,94,226,79,211)(65,225,110,210,95,195,80,240)(66,194,111,239,96,224,81,209)(67,223,112,208,97,193,82,238)(68,192,113,237,98,222,83,207)(69,221,114,206,99,191,84,236)(70,190,115,235,100,220,85,205)(71,219,116,204,101,189,86,234)(72,188,117,233,102,218,87,203)(73,217,118,202,103,187,88,232)(74,186,119,231,104,216,89,201)(75,215,120,200,105,185,90,230) );
G=PermutationGroup([[(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)], [(1,107,183,171,31,77,213,141),(2,96,184,160,32,66,214,130),(3,85,185,149,33,115,215,179),(4,74,186,138,34,104,216,168),(5,63,187,127,35,93,217,157),(6,112,188,176,36,82,218,146),(7,101,189,165,37,71,219,135),(8,90,190,154,38,120,220,124),(9,79,191,143,39,109,221,173),(10,68,192,132,40,98,222,162),(11,117,193,121,41,87,223,151),(12,106,194,170,42,76,224,140),(13,95,195,159,43,65,225,129),(14,84,196,148,44,114,226,178),(15,73,197,137,45,103,227,167),(16,62,198,126,46,92,228,156),(17,111,199,175,47,81,229,145),(18,100,200,164,48,70,230,134),(19,89,201,153,49,119,231,123),(20,78,202,142,50,108,232,172),(21,67,203,131,51,97,233,161),(22,116,204,180,52,86,234,150),(23,105,205,169,53,75,235,139),(24,94,206,158,54,64,236,128),(25,83,207,147,55,113,237,177),(26,72,208,136,56,102,238,166),(27,61,209,125,57,91,239,155),(28,110,210,174,58,80,240,144),(29,99,211,163,59,69,181,133),(30,88,212,152,60,118,182,122)], [(1,126,46,171,31,156,16,141),(2,155,47,140,32,125,17,170),(3,124,48,169,33,154,18,139),(4,153,49,138,34,123,19,168),(5,122,50,167,35,152,20,137),(6,151,51,136,36,121,21,166),(7,180,52,165,37,150,22,135),(8,149,53,134,38,179,23,164),(9,178,54,163,39,148,24,133),(10,147,55,132,40,177,25,162),(11,176,56,161,41,146,26,131),(12,145,57,130,42,175,27,160),(13,174,58,159,43,144,28,129),(14,143,59,128,44,173,29,158),(15,172,60,157,45,142,30,127),(61,229,106,214,91,199,76,184),(62,198,107,183,92,228,77,213),(63,227,108,212,93,197,78,182),(64,196,109,181,94,226,79,211),(65,225,110,210,95,195,80,240),(66,194,111,239,96,224,81,209),(67,223,112,208,97,193,82,238),(68,192,113,237,98,222,83,207),(69,221,114,206,99,191,84,236),(70,190,115,235,100,220,85,205),(71,219,116,204,101,189,86,234),(72,188,117,233,102,218,87,203),(73,217,118,202,103,187,88,232),(74,186,119,231,104,216,89,201),(75,215,120,200,105,185,90,230)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D5 | D6 | M4(2) | Dic5 | D10 | Dic5 | D12 | C3⋊D4 | C4×S3 | C5⋊D4 | C5⋊2C8 | S3×C8 | C8⋊S3 | C4.Dic5 | S3×D5 | C15⋊D4 | C5⋊D12 | S3×Dic5 | S3×C5⋊2C8 | D6.Dic5 |
| kernel | C60.94D4 | C6×C5⋊2C8 | C2×C15⋊3C8 | S3×C2×C20 | C10×Dic3 | S3×C2×C10 | S3×C10 | C2×C5⋊2C8 | C60 | S3×C2×C4 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C20 | C2×C10 | C12 | D6 | C10 | C10 | C6 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C60.94D4 ►in GL5(𝔽241)
| 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 |
| 0 | 64 | 64 | 0 | 0 |
| 0 | 0 | 0 | 216 | 0 |
| 0 | 0 | 0 | 152 | 135 |
| 233 | 0 | 0 | 0 | 0 |
| 0 | 163 | 85 | 0 | 0 |
| 0 | 156 | 78 | 0 | 0 |
| 0 | 0 | 0 | 123 | 28 |
| 0 | 0 | 0 | 228 | 118 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 163 | 85 | 0 | 0 |
| 0 | 163 | 78 | 0 | 0 |
| 0 | 0 | 0 | 123 | 28 |
| 0 | 0 | 0 | 189 | 118 |
G:=sub<GL(5,GF(241))| [64,0,0,0,0,0,0,64,0,0,0,177,64,0,0,0,0,0,216,152,0,0,0,0,135],[233,0,0,0,0,0,163,156,0,0,0,85,78,0,0,0,0,0,123,228,0,0,0,28,118],[8,0,0,0,0,0,163,163,0,0,0,85,78,0,0,0,0,0,123,189,0,0,0,28,118] >;
C60.94D4 in GAP, Magma, Sage, TeX
C_{60}._{94}D_4 % in TeX
G:=Group("C60.94D4"); // GroupNames label
G:=SmallGroup(480,32);
// by ID
G=gap.SmallGroup(480,32);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^49,c*a*c^-1=a^29,c*b*c^-1=a^15*b^3>;
// generators/relations