metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.39C23, Dic6.27D10, Dic10.15D6, D60.13C22, (Q8×D5)⋊6S3, C3⋊C8.10D10, C3⋊Q16⋊4D5, (C4×D5).12D6, (C6×D5).67D4, C15⋊Q16⋊7C2, C6.152(D4×D5), Q8.18(S3×D5), (C5×Q8).24D6, Q8⋊2D15⋊5C2, C3⋊6(Q16⋊D5), C30.201(C2×D4), C15⋊SD16⋊7C2, (C3×Q8).22D10, C12.28D10.1C2, C20.32D6⋊8C2, C5⋊3(Q8.11D6), C15⋊20(C8.C22), C20.39(C22×S3), (C3×Dic5).18D4, C12.39(C22×D5), (Q8×C15).9C22, D10.31(C3⋊D4), C15⋊3C8.13C22, (D5×C12).15C22, Dic5.25(C3⋊D4), (C5×Dic6).13C22, (C3×Dic10).14C22, (C3×Q8×D5)⋊3C2, C4.39(C2×S3×D5), (C5×C3⋊Q16)⋊5C2, C2.34(D5×C3⋊D4), C10.55(C2×C3⋊D4), (C5×C3⋊C8).13C22, SmallGroup(480,591)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.39C23
G = < a,b,c,d | a60=b2=1, c2=d2=a30, bab=a49, cac-1=a11, dad-1=a31, cbc-1=a30b, bd=db, dcd-1=a45c >
Subgroups: 652 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×Q8, C3×D5, D15, C30, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, C5×Q8, C5×Q8, C4.Dic3, Q8⋊2S3, C3⋊Q16, C3⋊Q16, C4○D12, C6×Q8, C5×Dic3, C3×Dic5, C3×Dic5, C60, C60, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, Q8.11D6, C5×C3⋊C8, C15⋊3C8, D30.C2, C3⋊D20, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C5×Dic6, D60, Q8×C15, Q16⋊D5, C20.32D6, C15⋊SD16, C15⋊Q16, C5×C3⋊Q16, Q8⋊2D15, C12.28D10, C3×Q8×D5, C60.39C23
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8.C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.11D6, C2×S3×D5, Q16⋊D5, D5×C3⋊D4, C60.39C23
►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)
(2 50)(3 39)(4 28)(5 17)(7 55)(8 44)(9 33)(10 22)(12 60)(13 49)(14 38)(15 27)(18 54)(19 43)(20 32)(23 59)(24 48)(25 37)(29 53)(30 42)(34 58)(35 47)(40 52)(45 57)(61 109)(62 98)(63 87)(64 76)(66 114)(67 103)(68 92)(69 81)(71 119)(72 108)(73 97)(74 86)(77 113)(78 102)(79 91)(82 118)(83 107)(84 96)(88 112)(89 101)(93 117)(94 106)(99 111)(104 116)(121 139)(122 128)(123 177)(124 166)(125 155)(126 144)(127 133)(129 171)(130 160)(131 149)(132 138)(134 176)(135 165)(136 154)(137 143)(140 170)(141 159)(142 148)(145 175)(146 164)(147 153)(150 180)(151 169)(152 158)(156 174)(157 163)(161 179)(162 168)(167 173)(172 178)(181 235)(182 224)(183 213)(184 202)(185 191)(186 240)(187 229)(188 218)(189 207)(190 196)(192 234)(193 223)(194 212)(195 201)(197 239)(198 228)(199 217)(200 206)(203 233)(204 222)(205 211)(208 238)(209 227)(210 216)(214 232)(215 221)(219 237)(220 226)(225 231)(230 236)
(1 180 31 150)(2 131 32 161)(3 142 33 172)(4 153 34 123)(5 164 35 134)(6 175 36 145)(7 126 37 156)(8 137 38 167)(9 148 39 178)(10 159 40 129)(11 170 41 140)(12 121 42 151)(13 132 43 162)(14 143 44 173)(15 154 45 124)(16 165 46 135)(17 176 47 146)(18 127 48 157)(19 138 49 168)(20 149 50 179)(21 160 51 130)(22 171 52 141)(23 122 53 152)(24 133 54 163)(25 144 55 174)(26 155 56 125)(27 166 57 136)(28 177 58 147)(29 128 59 158)(30 139 60 169)(61 234 91 204)(62 185 92 215)(63 196 93 226)(64 207 94 237)(65 218 95 188)(66 229 96 199)(67 240 97 210)(68 191 98 221)(69 202 99 232)(70 213 100 183)(71 224 101 194)(72 235 102 205)(73 186 103 216)(74 197 104 227)(75 208 105 238)(76 219 106 189)(77 230 107 200)(78 181 108 211)(79 192 109 222)(80 203 110 233)(81 214 111 184)(82 225 112 195)(83 236 113 206)(84 187 114 217)(85 198 115 228)(86 209 116 239)(87 220 117 190)(88 231 118 201)(89 182 119 212)(90 193 120 223)
(1 85 31 115)(2 116 32 86)(3 87 33 117)(4 118 34 88)(5 89 35 119)(6 120 36 90)(7 91 37 61)(8 62 38 92)(9 93 39 63)(10 64 40 94)(11 95 41 65)(12 66 42 96)(13 97 43 67)(14 68 44 98)(15 99 45 69)(16 70 46 100)(17 101 47 71)(18 72 48 102)(19 103 49 73)(20 74 50 104)(21 105 51 75)(22 76 52 106)(23 107 53 77)(24 78 54 108)(25 109 55 79)(26 80 56 110)(27 111 57 81)(28 82 58 112)(29 113 59 83)(30 84 60 114)(121 184 151 214)(122 215 152 185)(123 186 153 216)(124 217 154 187)(125 188 155 218)(126 219 156 189)(127 190 157 220)(128 221 158 191)(129 192 159 222)(130 223 160 193)(131 194 161 224)(132 225 162 195)(133 196 163 226)(134 227 164 197)(135 198 165 228)(136 229 166 199)(137 200 167 230)(138 231 168 201)(139 202 169 232)(140 233 170 203)(141 204 171 234)(142 235 172 205)(143 206 173 236)(144 237 174 207)(145 208 175 238)(146 239 176 209)(147 210 177 240)(148 181 178 211)(149 212 179 182)(150 183 180 213)
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), (2,50)(3,39)(4,28)(5,17)(7,55)(8,44)(9,33)(10,22)(12,60)(13,49)(14,38)(15,27)(18,54)(19,43)(20,32)(23,59)(24,48)(25,37)(29,53)(30,42)(34,58)(35,47)(40,52)(45,57)(61,109)(62,98)(63,87)(64,76)(66,114)(67,103)(68,92)(69,81)(71,119)(72,108)(73,97)(74,86)(77,113)(78,102)(79,91)(82,118)(83,107)(84,96)(88,112)(89,101)(93,117)(94,106)(99,111)(104,116)(121,139)(122,128)(123,177)(124,166)(125,155)(126,144)(127,133)(129,171)(130,160)(131,149)(132,138)(134,176)(135,165)(136,154)(137,143)(140,170)(141,159)(142,148)(145,175)(146,164)(147,153)(150,180)(151,169)(152,158)(156,174)(157,163)(161,179)(162,168)(167,173)(172,178)(181,235)(182,224)(183,213)(184,202)(185,191)(186,240)(187,229)(188,218)(189,207)(190,196)(192,234)(193,223)(194,212)(195,201)(197,239)(198,228)(199,217)(200,206)(203,233)(204,222)(205,211)(208,238)(209,227)(210,216)(214,232)(215,221)(219,237)(220,226)(225,231)(230,236), (1,180,31,150)(2,131,32,161)(3,142,33,172)(4,153,34,123)(5,164,35,134)(6,175,36,145)(7,126,37,156)(8,137,38,167)(9,148,39,178)(10,159,40,129)(11,170,41,140)(12,121,42,151)(13,132,43,162)(14,143,44,173)(15,154,45,124)(16,165,46,135)(17,176,47,146)(18,127,48,157)(19,138,49,168)(20,149,50,179)(21,160,51,130)(22,171,52,141)(23,122,53,152)(24,133,54,163)(25,144,55,174)(26,155,56,125)(27,166,57,136)(28,177,58,147)(29,128,59,158)(30,139,60,169)(61,234,91,204)(62,185,92,215)(63,196,93,226)(64,207,94,237)(65,218,95,188)(66,229,96,199)(67,240,97,210)(68,191,98,221)(69,202,99,232)(70,213,100,183)(71,224,101,194)(72,235,102,205)(73,186,103,216)(74,197,104,227)(75,208,105,238)(76,219,106,189)(77,230,107,200)(78,181,108,211)(79,192,109,222)(80,203,110,233)(81,214,111,184)(82,225,112,195)(83,236,113,206)(84,187,114,217)(85,198,115,228)(86,209,116,239)(87,220,117,190)(88,231,118,201)(89,182,119,212)(90,193,120,223), (1,85,31,115)(2,116,32,86)(3,87,33,117)(4,118,34,88)(5,89,35,119)(6,120,36,90)(7,91,37,61)(8,62,38,92)(9,93,39,63)(10,64,40,94)(11,95,41,65)(12,66,42,96)(13,97,43,67)(14,68,44,98)(15,99,45,69)(16,70,46,100)(17,101,47,71)(18,72,48,102)(19,103,49,73)(20,74,50,104)(21,105,51,75)(22,76,52,106)(23,107,53,77)(24,78,54,108)(25,109,55,79)(26,80,56,110)(27,111,57,81)(28,82,58,112)(29,113,59,83)(30,84,60,114)(121,184,151,214)(122,215,152,185)(123,186,153,216)(124,217,154,187)(125,188,155,218)(126,219,156,189)(127,190,157,220)(128,221,158,191)(129,192,159,222)(130,223,160,193)(131,194,161,224)(132,225,162,195)(133,196,163,226)(134,227,164,197)(135,198,165,228)(136,229,166,199)(137,200,167,230)(138,231,168,201)(139,202,169,232)(140,233,170,203)(141,204,171,234)(142,235,172,205)(143,206,173,236)(144,237,174,207)(145,208,175,238)(146,239,176,209)(147,210,177,240)(148,181,178,211)(149,212,179,182)(150,183,180,213)>;
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), (2,50)(3,39)(4,28)(5,17)(7,55)(8,44)(9,33)(10,22)(12,60)(13,49)(14,38)(15,27)(18,54)(19,43)(20,32)(23,59)(24,48)(25,37)(29,53)(30,42)(34,58)(35,47)(40,52)(45,57)(61,109)(62,98)(63,87)(64,76)(66,114)(67,103)(68,92)(69,81)(71,119)(72,108)(73,97)(74,86)(77,113)(78,102)(79,91)(82,118)(83,107)(84,96)(88,112)(89,101)(93,117)(94,106)(99,111)(104,116)(121,139)(122,128)(123,177)(124,166)(125,155)(126,144)(127,133)(129,171)(130,160)(131,149)(132,138)(134,176)(135,165)(136,154)(137,143)(140,170)(141,159)(142,148)(145,175)(146,164)(147,153)(150,180)(151,169)(152,158)(156,174)(157,163)(161,179)(162,168)(167,173)(172,178)(181,235)(182,224)(183,213)(184,202)(185,191)(186,240)(187,229)(188,218)(189,207)(190,196)(192,234)(193,223)(194,212)(195,201)(197,239)(198,228)(199,217)(200,206)(203,233)(204,222)(205,211)(208,238)(209,227)(210,216)(214,232)(215,221)(219,237)(220,226)(225,231)(230,236), (1,180,31,150)(2,131,32,161)(3,142,33,172)(4,153,34,123)(5,164,35,134)(6,175,36,145)(7,126,37,156)(8,137,38,167)(9,148,39,178)(10,159,40,129)(11,170,41,140)(12,121,42,151)(13,132,43,162)(14,143,44,173)(15,154,45,124)(16,165,46,135)(17,176,47,146)(18,127,48,157)(19,138,49,168)(20,149,50,179)(21,160,51,130)(22,171,52,141)(23,122,53,152)(24,133,54,163)(25,144,55,174)(26,155,56,125)(27,166,57,136)(28,177,58,147)(29,128,59,158)(30,139,60,169)(61,234,91,204)(62,185,92,215)(63,196,93,226)(64,207,94,237)(65,218,95,188)(66,229,96,199)(67,240,97,210)(68,191,98,221)(69,202,99,232)(70,213,100,183)(71,224,101,194)(72,235,102,205)(73,186,103,216)(74,197,104,227)(75,208,105,238)(76,219,106,189)(77,230,107,200)(78,181,108,211)(79,192,109,222)(80,203,110,233)(81,214,111,184)(82,225,112,195)(83,236,113,206)(84,187,114,217)(85,198,115,228)(86,209,116,239)(87,220,117,190)(88,231,118,201)(89,182,119,212)(90,193,120,223), (1,85,31,115)(2,116,32,86)(3,87,33,117)(4,118,34,88)(5,89,35,119)(6,120,36,90)(7,91,37,61)(8,62,38,92)(9,93,39,63)(10,64,40,94)(11,95,41,65)(12,66,42,96)(13,97,43,67)(14,68,44,98)(15,99,45,69)(16,70,46,100)(17,101,47,71)(18,72,48,102)(19,103,49,73)(20,74,50,104)(21,105,51,75)(22,76,52,106)(23,107,53,77)(24,78,54,108)(25,109,55,79)(26,80,56,110)(27,111,57,81)(28,82,58,112)(29,113,59,83)(30,84,60,114)(121,184,151,214)(122,215,152,185)(123,186,153,216)(124,217,154,187)(125,188,155,218)(126,219,156,189)(127,190,157,220)(128,221,158,191)(129,192,159,222)(130,223,160,193)(131,194,161,224)(132,225,162,195)(133,196,163,226)(134,227,164,197)(135,198,165,228)(136,229,166,199)(137,200,167,230)(138,231,168,201)(139,202,169,232)(140,233,170,203)(141,204,171,234)(142,235,172,205)(143,206,173,236)(144,237,174,207)(145,208,175,238)(146,239,176,209)(147,210,177,240)(148,181,178,211)(149,212,179,182)(150,183,180,213) );
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)], [(2,50),(3,39),(4,28),(5,17),(7,55),(8,44),(9,33),(10,22),(12,60),(13,49),(14,38),(15,27),(18,54),(19,43),(20,32),(23,59),(24,48),(25,37),(29,53),(30,42),(34,58),(35,47),(40,52),(45,57),(61,109),(62,98),(63,87),(64,76),(66,114),(67,103),(68,92),(69,81),(71,119),(72,108),(73,97),(74,86),(77,113),(78,102),(79,91),(82,118),(83,107),(84,96),(88,112),(89,101),(93,117),(94,106),(99,111),(104,116),(121,139),(122,128),(123,177),(124,166),(125,155),(126,144),(127,133),(129,171),(130,160),(131,149),(132,138),(134,176),(135,165),(136,154),(137,143),(140,170),(141,159),(142,148),(145,175),(146,164),(147,153),(150,180),(151,169),(152,158),(156,174),(157,163),(161,179),(162,168),(167,173),(172,178),(181,235),(182,224),(183,213),(184,202),(185,191),(186,240),(187,229),(188,218),(189,207),(190,196),(192,234),(193,223),(194,212),(195,201),(197,239),(198,228),(199,217),(200,206),(203,233),(204,222),(205,211),(208,238),(209,227),(210,216),(214,232),(215,221),(219,237),(220,226),(225,231),(230,236)], [(1,180,31,150),(2,131,32,161),(3,142,33,172),(4,153,34,123),(5,164,35,134),(6,175,36,145),(7,126,37,156),(8,137,38,167),(9,148,39,178),(10,159,40,129),(11,170,41,140),(12,121,42,151),(13,132,43,162),(14,143,44,173),(15,154,45,124),(16,165,46,135),(17,176,47,146),(18,127,48,157),(19,138,49,168),(20,149,50,179),(21,160,51,130),(22,171,52,141),(23,122,53,152),(24,133,54,163),(25,144,55,174),(26,155,56,125),(27,166,57,136),(28,177,58,147),(29,128,59,158),(30,139,60,169),(61,234,91,204),(62,185,92,215),(63,196,93,226),(64,207,94,237),(65,218,95,188),(66,229,96,199),(67,240,97,210),(68,191,98,221),(69,202,99,232),(70,213,100,183),(71,224,101,194),(72,235,102,205),(73,186,103,216),(74,197,104,227),(75,208,105,238),(76,219,106,189),(77,230,107,200),(78,181,108,211),(79,192,109,222),(80,203,110,233),(81,214,111,184),(82,225,112,195),(83,236,113,206),(84,187,114,217),(85,198,115,228),(86,209,116,239),(87,220,117,190),(88,231,118,201),(89,182,119,212),(90,193,120,223)], [(1,85,31,115),(2,116,32,86),(3,87,33,117),(4,118,34,88),(5,89,35,119),(6,120,36,90),(7,91,37,61),(8,62,38,92),(9,93,39,63),(10,64,40,94),(11,95,41,65),(12,66,42,96),(13,97,43,67),(14,68,44,98),(15,99,45,69),(16,70,46,100),(17,101,47,71),(18,72,48,102),(19,103,49,73),(20,74,50,104),(21,105,51,75),(22,76,52,106),(23,107,53,77),(24,78,54,108),(25,109,55,79),(26,80,56,110),(27,111,57,81),(28,82,58,112),(29,113,59,83),(30,84,60,114),(121,184,151,214),(122,215,152,185),(123,186,153,216),(124,217,154,187),(125,188,155,218),(126,219,156,189),(127,190,157,220),(128,221,158,191),(129,192,159,222),(130,223,160,193),(131,194,161,224),(132,225,162,195),(133,196,163,226),(134,227,164,197),(135,198,165,228),(136,229,166,199),(137,200,167,230),(138,231,168,201),(139,202,169,232),(140,233,170,203),(141,204,171,234),(142,235,172,205),(143,206,173,236),(144,237,174,207),(145,208,175,238),(146,239,176,209),(147,210,177,240),(148,181,178,211),(149,212,179,182),(150,183,180,213)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 10 | 60 | 2 | 2 | 4 | 10 | 12 | 20 | 2 | 2 | 2 | 10 | 10 | 12 | 60 | 2 | 2 | 4 | 4 | 4 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 8 | 8 | 24 | 24 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | C3⋊D4 | C3⋊D4 | C8.C22 | S3×D5 | D4×D5 | Q8.11D6 | C2×S3×D5 | Q16⋊D5 | D5×C3⋊D4 | C60.39C23 |
| kernel | C60.39C23 | C20.32D6 | C15⋊SD16 | C15⋊Q16 | C5×C3⋊Q16 | Q8⋊2D15 | C12.28D10 | C3×Q8×D5 | Q8×D5 | C3×Dic5 | C6×D5 | C3⋊Q16 | Dic10 | C4×D5 | C5×Q8 | C3⋊C8 | Dic6 | C3×Q8 | Dic5 | D10 | C15 | 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 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of C60.39C23 ►in GL8(𝔽241)
| 1 | 189 | 0 | 0 | 0 | 0 | 0 | 0 |
| 52 | 189 | 0 | 0 | 0 | 0 | 0 | 0 |
| 97 | 0 | 190 | 189 | 0 | 0 | 0 | 0 |
| 127 | 144 | 51 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 226 | 188 | 71 | 209 |
| 0 | 0 | 0 | 0 | 154 | 15 | 89 | 89 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 225 | 0 |
| 189 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 144 | 0 | 51 | 52 | 0 | 0 | 0 | 0 |
| 114 | 97 | 191 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 142 | 142 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 145 | 0 | 180 | 219 | 0 | 0 | 0 | 0 |
| 98 | 2 | 44 | 22 | 0 | 0 | 0 | 0 |
| 227 | 227 | 239 | 0 | 0 | 0 | 0 | 0 |
| 140 | 236 | 206 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 219 | 0 | 67 | 67 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 187 | 1 | 22 | 22 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 1 | 0 | 104 | 239 | 0 | 0 | 0 | 0 |
| 228 | 229 | 4 | 2 | 0 | 0 | 0 | 0 |
| 50 | 50 | 12 | 0 | 0 | 0 | 0 | 0 |
| 190 | 190 | 194 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 141 | 87 | 227 | 19 |
| 0 | 0 | 0 | 0 | 223 | 100 | 168 | 168 |
| 0 | 0 | 0 | 0 | 0 | 0 | 80 | 180 |
| 0 | 0 | 0 | 0 | 0 | 0 | 180 | 161 |
G:=sub<GL(8,GF(241))| [1,52,97,127,0,0,0,0,189,189,0,144,0,0,0,0,0,0,190,51,0,0,0,0,0,0,189,0,0,0,0,0,0,0,0,0,226,154,0,0,0,0,0,0,188,15,0,0,0,0,0,0,71,89,0,225,0,0,0,0,209,89,16,0],[189,240,144,114,0,0,0,0,52,52,0,97,0,0,0,0,0,0,51,191,0,0,0,0,0,0,52,190,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,142,0,240,0,0,0,0,0,142,0,0,240],[145,98,227,140,0,0,0,0,0,2,227,236,0,0,0,0,180,44,239,206,0,0,0,0,219,22,0,96,0,0,0,0,0,0,0,0,219,0,187,0,0,0,0,0,0,0,1,240,0,0,0,0,67,0,22,0,0,0,0,0,67,1,22,0],[1,228,50,190,0,0,0,0,0,229,50,190,0,0,0,0,104,4,12,194,0,0,0,0,239,2,0,240,0,0,0,0,0,0,0,0,141,223,0,0,0,0,0,0,87,100,0,0,0,0,0,0,227,168,80,180,0,0,0,0,19,168,180,161] >;
C60.39C23 in GAP, Magma, Sage, TeX
C_{60}._{39}C_2^3 % in TeX
G:=Group("C60.39C2^3"); // GroupNames label
G:=SmallGroup(480,591);
// by ID
G=gap.SmallGroup(480,591);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,135,100,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^60=b^2=1,c^2=d^2=a^30,b*a*b=a^49,c*a*c^-1=a^11,d*a*d^-1=a^31,c*b*c^-1=a^30*b,b*d=d*b,d*c*d^-1=a^45*c>;
// generators/relations