metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.13D4, C60.36C23, Dic6.13D10, Dic15.45D4, Dic10.13D6, D60.12C22, C3⋊C8.8D10, D15⋊Q8⋊3C2, C5⋊Q16⋊3S3, C3⋊Q16⋊3D5, C6.78(D4×D5), C5⋊2C8.8D6, C10.79(S3×D4), C5⋊4(Q16⋊S3), (C3×Q8).4D10, Q8.21(S3×D5), (C5×Q8).21D6, C3⋊4(Q16⋊D5), C15⋊SD16⋊6C2, Dic6⋊D5⋊6C2, C30.198(C2×D4), D30.5C4⋊6C2, Q8⋊3D15.1C2, C15⋊18(C8.C22), C20.36(C22×S3), C12.36(C22×D5), (Q8×C15).6C22, (C4×D15).12C22, C2.31(D10⋊D6), (C5×Dic6).12C22, (C3×Dic10).12C22, C4.36(C2×S3×D5), (C3×C5⋊Q16)⋊4C2, (C5×C3⋊Q16)⋊4C2, (C5×C3⋊C8).10C22, (C3×C5⋊2C8).10C22, SmallGroup(480,588)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.C23
G = < a,b,c,d | a60=1, b2=c2=d2=a30, bab-1=a19, cac-1=a11, dad-1=a31, bc=cb, dbd-1=a45b, dcd-1=a15c >
Subgroups: 748 in 120 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, C12, D6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C3×Q8, C3×Q8, D15, C30, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, C5×Q8, C5×Q8, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, C5×Dic3, C3×Dic5, Dic15, C60, C60, D30, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, Q16⋊S3, C5×C3⋊C8, C3×C5⋊2C8, D30.C2, C15⋊Q8, C3×Dic10, C5×Dic6, C4×D15, C4×D15, D60, D60, Q8×C15, Q16⋊D5, D30.5C4, C15⋊SD16, Dic6⋊D5, C3×C5⋊Q16, C5×C3⋊Q16, D15⋊Q8, Q8⋊3D15, C60.C23
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, C22×D5, S3×D4, S3×D5, D4×D5, Q16⋊S3, C2×S3×D5, Q16⋊D5, D10⋊D6, C60.C23
►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)
(1 170 31 140)(2 129 32 159)(3 148 33 178)(4 167 34 137)(5 126 35 156)(6 145 36 175)(7 164 37 134)(8 123 38 153)(9 142 39 172)(10 161 40 131)(11 180 41 150)(12 139 42 169)(13 158 43 128)(14 177 44 147)(15 136 45 166)(16 155 46 125)(17 174 47 144)(18 133 48 163)(19 152 49 122)(20 171 50 141)(21 130 51 160)(22 149 52 179)(23 168 53 138)(24 127 54 157)(25 146 55 176)(26 165 56 135)(27 124 57 154)(28 143 58 173)(29 162 59 132)(30 121 60 151)(61 217 91 187)(62 236 92 206)(63 195 93 225)(64 214 94 184)(65 233 95 203)(66 192 96 222)(67 211 97 181)(68 230 98 200)(69 189 99 219)(70 208 100 238)(71 227 101 197)(72 186 102 216)(73 205 103 235)(74 224 104 194)(75 183 105 213)(76 202 106 232)(77 221 107 191)(78 240 108 210)(79 199 109 229)(80 218 110 188)(81 237 111 207)(82 196 112 226)(83 215 113 185)(84 234 114 204)(85 193 115 223)(86 212 116 182)(87 231 117 201)(88 190 118 220)(89 209 119 239)(90 228 120 198)
(1 140 31 170)(2 151 32 121)(3 162 33 132)(4 173 34 143)(5 124 35 154)(6 135 36 165)(7 146 37 176)(8 157 38 127)(9 168 39 138)(10 179 40 149)(11 130 41 160)(12 141 42 171)(13 152 43 122)(14 163 44 133)(15 174 45 144)(16 125 46 155)(17 136 47 166)(18 147 48 177)(19 158 49 128)(20 169 50 139)(21 180 51 150)(22 131 52 161)(23 142 53 172)(24 153 54 123)(25 164 55 134)(26 175 56 145)(27 126 57 156)(28 137 58 167)(29 148 59 178)(30 159 60 129)(61 213 91 183)(62 224 92 194)(63 235 93 205)(64 186 94 216)(65 197 95 227)(66 208 96 238)(67 219 97 189)(68 230 98 200)(69 181 99 211)(70 192 100 222)(71 203 101 233)(72 214 102 184)(73 225 103 195)(74 236 104 206)(75 187 105 217)(76 198 106 228)(77 209 107 239)(78 220 108 190)(79 231 109 201)(80 182 110 212)(81 193 111 223)(82 204 112 234)(83 215 113 185)(84 226 114 196)(85 237 115 207)(86 188 116 218)(87 199 117 229)(88 210 118 240)(89 221 119 191)(90 232 120 202)
(1 98 31 68)(2 69 32 99)(3 100 33 70)(4 71 34 101)(5 102 35 72)(6 73 36 103)(7 104 37 74)(8 75 38 105)(9 106 39 76)(10 77 40 107)(11 108 41 78)(12 79 42 109)(13 110 43 80)(14 81 44 111)(15 112 45 82)(16 83 46 113)(17 114 47 84)(18 85 48 115)(19 116 49 86)(20 87 50 117)(21 118 51 88)(22 89 52 119)(23 120 53 90)(24 91 54 61)(25 62 55 92)(26 93 56 63)(27 64 57 94)(28 95 58 65)(29 66 59 96)(30 97 60 67)(121 196 151 226)(122 227 152 197)(123 198 153 228)(124 229 154 199)(125 200 155 230)(126 231 156 201)(127 202 157 232)(128 233 158 203)(129 204 159 234)(130 235 160 205)(131 206 161 236)(132 237 162 207)(133 208 163 238)(134 239 164 209)(135 210 165 240)(136 181 166 211)(137 212 167 182)(138 183 168 213)(139 214 169 184)(140 185 170 215)(141 216 171 186)(142 187 172 217)(143 218 173 188)(144 189 174 219)(145 220 175 190)(146 191 176 221)(147 222 177 192)(148 193 178 223)(149 224 179 194)(150 195 180 225)
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), (1,170,31,140)(2,129,32,159)(3,148,33,178)(4,167,34,137)(5,126,35,156)(6,145,36,175)(7,164,37,134)(8,123,38,153)(9,142,39,172)(10,161,40,131)(11,180,41,150)(12,139,42,169)(13,158,43,128)(14,177,44,147)(15,136,45,166)(16,155,46,125)(17,174,47,144)(18,133,48,163)(19,152,49,122)(20,171,50,141)(21,130,51,160)(22,149,52,179)(23,168,53,138)(24,127,54,157)(25,146,55,176)(26,165,56,135)(27,124,57,154)(28,143,58,173)(29,162,59,132)(30,121,60,151)(61,217,91,187)(62,236,92,206)(63,195,93,225)(64,214,94,184)(65,233,95,203)(66,192,96,222)(67,211,97,181)(68,230,98,200)(69,189,99,219)(70,208,100,238)(71,227,101,197)(72,186,102,216)(73,205,103,235)(74,224,104,194)(75,183,105,213)(76,202,106,232)(77,221,107,191)(78,240,108,210)(79,199,109,229)(80,218,110,188)(81,237,111,207)(82,196,112,226)(83,215,113,185)(84,234,114,204)(85,193,115,223)(86,212,116,182)(87,231,117,201)(88,190,118,220)(89,209,119,239)(90,228,120,198), (1,140,31,170)(2,151,32,121)(3,162,33,132)(4,173,34,143)(5,124,35,154)(6,135,36,165)(7,146,37,176)(8,157,38,127)(9,168,39,138)(10,179,40,149)(11,130,41,160)(12,141,42,171)(13,152,43,122)(14,163,44,133)(15,174,45,144)(16,125,46,155)(17,136,47,166)(18,147,48,177)(19,158,49,128)(20,169,50,139)(21,180,51,150)(22,131,52,161)(23,142,53,172)(24,153,54,123)(25,164,55,134)(26,175,56,145)(27,126,57,156)(28,137,58,167)(29,148,59,178)(30,159,60,129)(61,213,91,183)(62,224,92,194)(63,235,93,205)(64,186,94,216)(65,197,95,227)(66,208,96,238)(67,219,97,189)(68,230,98,200)(69,181,99,211)(70,192,100,222)(71,203,101,233)(72,214,102,184)(73,225,103,195)(74,236,104,206)(75,187,105,217)(76,198,106,228)(77,209,107,239)(78,220,108,190)(79,231,109,201)(80,182,110,212)(81,193,111,223)(82,204,112,234)(83,215,113,185)(84,226,114,196)(85,237,115,207)(86,188,116,218)(87,199,117,229)(88,210,118,240)(89,221,119,191)(90,232,120,202), (1,98,31,68)(2,69,32,99)(3,100,33,70)(4,71,34,101)(5,102,35,72)(6,73,36,103)(7,104,37,74)(8,75,38,105)(9,106,39,76)(10,77,40,107)(11,108,41,78)(12,79,42,109)(13,110,43,80)(14,81,44,111)(15,112,45,82)(16,83,46,113)(17,114,47,84)(18,85,48,115)(19,116,49,86)(20,87,50,117)(21,118,51,88)(22,89,52,119)(23,120,53,90)(24,91,54,61)(25,62,55,92)(26,93,56,63)(27,64,57,94)(28,95,58,65)(29,66,59,96)(30,97,60,67)(121,196,151,226)(122,227,152,197)(123,198,153,228)(124,229,154,199)(125,200,155,230)(126,231,156,201)(127,202,157,232)(128,233,158,203)(129,204,159,234)(130,235,160,205)(131,206,161,236)(132,237,162,207)(133,208,163,238)(134,239,164,209)(135,210,165,240)(136,181,166,211)(137,212,167,182)(138,183,168,213)(139,214,169,184)(140,185,170,215)(141,216,171,186)(142,187,172,217)(143,218,173,188)(144,189,174,219)(145,220,175,190)(146,191,176,221)(147,222,177,192)(148,193,178,223)(149,224,179,194)(150,195,180,225)>;
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), (1,170,31,140)(2,129,32,159)(3,148,33,178)(4,167,34,137)(5,126,35,156)(6,145,36,175)(7,164,37,134)(8,123,38,153)(9,142,39,172)(10,161,40,131)(11,180,41,150)(12,139,42,169)(13,158,43,128)(14,177,44,147)(15,136,45,166)(16,155,46,125)(17,174,47,144)(18,133,48,163)(19,152,49,122)(20,171,50,141)(21,130,51,160)(22,149,52,179)(23,168,53,138)(24,127,54,157)(25,146,55,176)(26,165,56,135)(27,124,57,154)(28,143,58,173)(29,162,59,132)(30,121,60,151)(61,217,91,187)(62,236,92,206)(63,195,93,225)(64,214,94,184)(65,233,95,203)(66,192,96,222)(67,211,97,181)(68,230,98,200)(69,189,99,219)(70,208,100,238)(71,227,101,197)(72,186,102,216)(73,205,103,235)(74,224,104,194)(75,183,105,213)(76,202,106,232)(77,221,107,191)(78,240,108,210)(79,199,109,229)(80,218,110,188)(81,237,111,207)(82,196,112,226)(83,215,113,185)(84,234,114,204)(85,193,115,223)(86,212,116,182)(87,231,117,201)(88,190,118,220)(89,209,119,239)(90,228,120,198), (1,140,31,170)(2,151,32,121)(3,162,33,132)(4,173,34,143)(5,124,35,154)(6,135,36,165)(7,146,37,176)(8,157,38,127)(9,168,39,138)(10,179,40,149)(11,130,41,160)(12,141,42,171)(13,152,43,122)(14,163,44,133)(15,174,45,144)(16,125,46,155)(17,136,47,166)(18,147,48,177)(19,158,49,128)(20,169,50,139)(21,180,51,150)(22,131,52,161)(23,142,53,172)(24,153,54,123)(25,164,55,134)(26,175,56,145)(27,126,57,156)(28,137,58,167)(29,148,59,178)(30,159,60,129)(61,213,91,183)(62,224,92,194)(63,235,93,205)(64,186,94,216)(65,197,95,227)(66,208,96,238)(67,219,97,189)(68,230,98,200)(69,181,99,211)(70,192,100,222)(71,203,101,233)(72,214,102,184)(73,225,103,195)(74,236,104,206)(75,187,105,217)(76,198,106,228)(77,209,107,239)(78,220,108,190)(79,231,109,201)(80,182,110,212)(81,193,111,223)(82,204,112,234)(83,215,113,185)(84,226,114,196)(85,237,115,207)(86,188,116,218)(87,199,117,229)(88,210,118,240)(89,221,119,191)(90,232,120,202), (1,98,31,68)(2,69,32,99)(3,100,33,70)(4,71,34,101)(5,102,35,72)(6,73,36,103)(7,104,37,74)(8,75,38,105)(9,106,39,76)(10,77,40,107)(11,108,41,78)(12,79,42,109)(13,110,43,80)(14,81,44,111)(15,112,45,82)(16,83,46,113)(17,114,47,84)(18,85,48,115)(19,116,49,86)(20,87,50,117)(21,118,51,88)(22,89,52,119)(23,120,53,90)(24,91,54,61)(25,62,55,92)(26,93,56,63)(27,64,57,94)(28,95,58,65)(29,66,59,96)(30,97,60,67)(121,196,151,226)(122,227,152,197)(123,198,153,228)(124,229,154,199)(125,200,155,230)(126,231,156,201)(127,202,157,232)(128,233,158,203)(129,204,159,234)(130,235,160,205)(131,206,161,236)(132,237,162,207)(133,208,163,238)(134,239,164,209)(135,210,165,240)(136,181,166,211)(137,212,167,182)(138,183,168,213)(139,214,169,184)(140,185,170,215)(141,216,171,186)(142,187,172,217)(143,218,173,188)(144,189,174,219)(145,220,175,190)(146,191,176,221)(147,222,177,192)(148,193,178,223)(149,224,179,194)(150,195,180,225) );
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)], [(1,170,31,140),(2,129,32,159),(3,148,33,178),(4,167,34,137),(5,126,35,156),(6,145,36,175),(7,164,37,134),(8,123,38,153),(9,142,39,172),(10,161,40,131),(11,180,41,150),(12,139,42,169),(13,158,43,128),(14,177,44,147),(15,136,45,166),(16,155,46,125),(17,174,47,144),(18,133,48,163),(19,152,49,122),(20,171,50,141),(21,130,51,160),(22,149,52,179),(23,168,53,138),(24,127,54,157),(25,146,55,176),(26,165,56,135),(27,124,57,154),(28,143,58,173),(29,162,59,132),(30,121,60,151),(61,217,91,187),(62,236,92,206),(63,195,93,225),(64,214,94,184),(65,233,95,203),(66,192,96,222),(67,211,97,181),(68,230,98,200),(69,189,99,219),(70,208,100,238),(71,227,101,197),(72,186,102,216),(73,205,103,235),(74,224,104,194),(75,183,105,213),(76,202,106,232),(77,221,107,191),(78,240,108,210),(79,199,109,229),(80,218,110,188),(81,237,111,207),(82,196,112,226),(83,215,113,185),(84,234,114,204),(85,193,115,223),(86,212,116,182),(87,231,117,201),(88,190,118,220),(89,209,119,239),(90,228,120,198)], [(1,140,31,170),(2,151,32,121),(3,162,33,132),(4,173,34,143),(5,124,35,154),(6,135,36,165),(7,146,37,176),(8,157,38,127),(9,168,39,138),(10,179,40,149),(11,130,41,160),(12,141,42,171),(13,152,43,122),(14,163,44,133),(15,174,45,144),(16,125,46,155),(17,136,47,166),(18,147,48,177),(19,158,49,128),(20,169,50,139),(21,180,51,150),(22,131,52,161),(23,142,53,172),(24,153,54,123),(25,164,55,134),(26,175,56,145),(27,126,57,156),(28,137,58,167),(29,148,59,178),(30,159,60,129),(61,213,91,183),(62,224,92,194),(63,235,93,205),(64,186,94,216),(65,197,95,227),(66,208,96,238),(67,219,97,189),(68,230,98,200),(69,181,99,211),(70,192,100,222),(71,203,101,233),(72,214,102,184),(73,225,103,195),(74,236,104,206),(75,187,105,217),(76,198,106,228),(77,209,107,239),(78,220,108,190),(79,231,109,201),(80,182,110,212),(81,193,111,223),(82,204,112,234),(83,215,113,185),(84,226,114,196),(85,237,115,207),(86,188,116,218),(87,199,117,229),(88,210,118,240),(89,221,119,191),(90,232,120,202)], [(1,98,31,68),(2,69,32,99),(3,100,33,70),(4,71,34,101),(5,102,35,72),(6,73,36,103),(7,104,37,74),(8,75,38,105),(9,106,39,76),(10,77,40,107),(11,108,41,78),(12,79,42,109),(13,110,43,80),(14,81,44,111),(15,112,45,82),(16,83,46,113),(17,114,47,84),(18,85,48,115),(19,116,49,86),(20,87,50,117),(21,118,51,88),(22,89,52,119),(23,120,53,90),(24,91,54,61),(25,62,55,92),(26,93,56,63),(27,64,57,94),(28,95,58,65),(29,66,59,96),(30,97,60,67),(121,196,151,226),(122,227,152,197),(123,198,153,228),(124,229,154,199),(125,200,155,230),(126,231,156,201),(127,202,157,232),(128,233,158,203),(129,204,159,234),(130,235,160,205),(131,206,161,236),(132,237,162,207),(133,208,163,238),(134,239,164,209),(135,210,165,240),(136,181,166,211),(137,212,167,182),(138,183,168,213),(139,214,169,184),(140,185,170,215),(141,216,171,186),(142,187,172,217),(143,218,173,188),(144,189,174,219),(145,220,175,190),(146,191,176,221),(147,222,177,192),(148,193,178,223),(149,224,179,194),(150,195,180,225)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 30 | 60 | 2 | 2 | 4 | 12 | 20 | 30 | 2 | 2 | 2 | 12 | 20 | 2 | 2 | 4 | 8 | 40 | 4 | 4 | 4 | 4 | 8 | 8 | 24 | 24 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C8.C22 | S3×D4 | S3×D5 | D4×D5 | Q16⋊S3 | C2×S3×D5 | Q16⋊D5 | D10⋊D6 | C60.C23 |
| kernel | C60.C23 | D30.5C4 | C15⋊SD16 | Dic6⋊D5 | C3×C5⋊Q16 | C5×C3⋊Q16 | D15⋊Q8 | Q8⋊3D15 | C5⋊Q16 | Dic15 | D30 | C3⋊Q16 | C5⋊2C8 | Dic10 | C5×Q8 | C3⋊C8 | Dic6 | C3×Q8 | C15 | C10 | Q8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of C60.C23 ►in GL6(𝔽241)
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 52 | 137 | 137 |
| 0 | 0 | 189 | 240 | 104 | 2 |
| 0 | 0 | 52 | 52 | 189 | 189 |
| 0 | 0 | 189 | 240 | 52 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 120 | 0 | 25 | 0 |
| 0 | 0 | 26 | 121 | 146 | 216 |
| 0 | 0 | 12 | 0 | 121 | 0 |
| 0 | 0 | 99 | 229 | 215 | 120 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 121 | 0 | 216 | 0 |
| 0 | 0 | 0 | 121 | 0 | 216 |
| 0 | 0 | 229 | 0 | 120 | 0 |
| 0 | 0 | 0 | 229 | 0 | 120 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 195 | 128 | 148 | 154 |
| 0 | 0 | 113 | 46 | 87 | 93 |
| 0 | 0 | 28 | 205 | 46 | 113 |
| 0 | 0 | 36 | 213 | 128 | 195 |
G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,240,0,0,0,0,0,0,52,189,52,189,0,0,52,240,52,240,0,0,137,104,189,52,0,0,137,2,189,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,120,26,12,99,0,0,0,121,0,229,0,0,25,146,121,215,0,0,0,216,0,120],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,121,0,229,0,0,0,0,121,0,229,0,0,216,0,120,0,0,0,0,216,0,120],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,195,113,28,36,0,0,128,46,205,213,0,0,148,87,46,128,0,0,154,93,113,195] >;
C60.C23 in GAP, Magma, Sage, TeX
C_{60}.C_2^3 % in TeX
G:=Group("C60.C2^3"); // GroupNames label
G:=SmallGroup(480,588);
// by ID
G=gap.SmallGroup(480,588);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,303,100,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^60=1,b^2=c^2=d^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^11,d*a*d^-1=a^31,b*c=c*b,d*b*d^-1=a^45*b,d*c*d^-1=a^15*c>;
// generators/relations