metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.8C23, Dic10.6D6, Dic6.23D10, Dic30.1C22, C3⋊C8.5D10, D4.S3⋊1D5, D4.8(S3×D5), (C6×D5).9D4, (C4×D5).6D6, (C5×D4).2D6, D4.D15⋊2C2, (D5×Dic6)⋊2C2, C15⋊Q16⋊1C2, C3⋊Dic20⋊1C2, C6.141(D4×D5), D4⋊2D5.1S3, (C3×D4).19D10, C30.170(C2×D4), C5⋊2(Q8.14D6), C3⋊6(SD16⋊D5), C20.8(C22×S3), C12.8(C22×D5), C20.32D6⋊2C2, C15⋊13(C8.C22), C15⋊3C8.1C22, (C3×Dic5).67D4, (D4×C15).2C22, (D5×C12).4C22, D10.18(C3⋊D4), (C5×Dic6).1C22, Dic5.32(C3⋊D4), (C3×Dic10).1C22, C4.8(C2×S3×D5), (C5×D4.S3)⋊2C2, C2.23(D5×C3⋊D4), (C5×C3⋊C8).1C22, C10.44(C2×C3⋊D4), (C3×D4⋊2D5).1C2, SmallGroup(480,560)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.8C23
G = < a,b,c,d | a60=b2=d2=1, c2=a30, bab=a49, cac-1=a11, dad=a31, bc=cb, dbd=a30b, dcd=a45c >
Subgroups: 572 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C4.Dic3, D4.S3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C6×D5, C2×C30, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8×D5, Q8.14D6, C5×C3⋊C8, C15⋊3C8, D5×Dic3, C15⋊Q8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×Dic6, Dic30, D4×C15, SD16⋊D5, C20.32D6, C15⋊Q16, C3⋊Dic20, C5×D4.S3, D4.D15, D5×Dic6, C3×D4⋊2D5, C60.8C23
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, Q8.14D6, C2×S3×D5, SD16⋊D5, D5×C3⋊D4, C60.8C23
►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)(61 103)(62 92)(63 81)(64 70)(65 119)(66 108)(67 97)(68 86)(69 75)(71 113)(72 102)(73 91)(74 80)(76 118)(77 107)(78 96)(79 85)(82 112)(83 101)(84 90)(87 117)(88 106)(89 95)(93 111)(94 100)(98 116)(99 105)(104 110)(109 115)(114 120)(121 169)(122 158)(123 147)(124 136)(126 174)(127 163)(128 152)(129 141)(131 179)(132 168)(133 157)(134 146)(137 173)(138 162)(139 151)(142 178)(143 167)(144 156)(148 172)(149 161)(153 177)(154 166)(159 171)(164 176)(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)
(1 165 31 135)(2 176 32 146)(3 127 33 157)(4 138 34 168)(5 149 35 179)(6 160 36 130)(7 171 37 141)(8 122 38 152)(9 133 39 163)(10 144 40 174)(11 155 41 125)(12 166 42 136)(13 177 43 147)(14 128 44 158)(15 139 45 169)(16 150 46 180)(17 161 47 131)(18 172 48 142)(19 123 49 153)(20 134 50 164)(21 145 51 175)(22 156 52 126)(23 167 53 137)(24 178 54 148)(25 129 55 159)(26 140 56 170)(27 151 57 121)(28 162 58 132)(29 173 59 143)(30 124 60 154)(61 182 91 212)(62 193 92 223)(63 204 93 234)(64 215 94 185)(65 226 95 196)(66 237 96 207)(67 188 97 218)(68 199 98 229)(69 210 99 240)(70 221 100 191)(71 232 101 202)(72 183 102 213)(73 194 103 224)(74 205 104 235)(75 216 105 186)(76 227 106 197)(77 238 107 208)(78 189 108 219)(79 200 109 230)(80 211 110 181)(81 222 111 192)(82 233 112 203)(83 184 113 214)(84 195 114 225)(85 206 115 236)(86 217 116 187)(87 228 117 198)(88 239 118 209)(89 190 119 220)(90 201 120 231)
(1 102)(2 73)(3 104)(4 75)(5 106)(6 77)(7 108)(8 79)(9 110)(10 81)(11 112)(12 83)(13 114)(14 85)(15 116)(16 87)(17 118)(18 89)(19 120)(20 91)(21 62)(22 93)(23 64)(24 95)(25 66)(26 97)(27 68)(28 99)(29 70)(30 101)(31 72)(32 103)(33 74)(34 105)(35 76)(36 107)(37 78)(38 109)(39 80)(40 111)(41 82)(42 113)(43 84)(44 115)(45 86)(46 117)(47 88)(48 119)(49 90)(50 61)(51 92)(52 63)(53 94)(54 65)(55 96)(56 67)(57 98)(58 69)(59 100)(60 71)(121 184)(122 215)(123 186)(124 217)(125 188)(126 219)(127 190)(128 221)(129 192)(130 223)(131 194)(132 225)(133 196)(134 227)(135 198)(136 229)(137 200)(138 231)(139 202)(140 233)(141 204)(142 235)(143 206)(144 237)(145 208)(146 239)(147 210)(148 181)(149 212)(150 183)(151 214)(152 185)(153 216)(154 187)(155 218)(156 189)(157 220)(158 191)(159 222)(160 193)(161 224)(162 195)(163 226)(164 197)(165 228)(166 199)(167 230)(168 201)(169 232)(170 203)(171 234)(172 205)(173 236)(174 207)(175 238)(176 209)(177 240)(178 211)(179 182)(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)(61,103)(62,92)(63,81)(64,70)(65,119)(66,108)(67,97)(68,86)(69,75)(71,113)(72,102)(73,91)(74,80)(76,118)(77,107)(78,96)(79,85)(82,112)(83,101)(84,90)(87,117)(88,106)(89,95)(93,111)(94,100)(98,116)(99,105)(104,110)(109,115)(114,120)(121,169)(122,158)(123,147)(124,136)(126,174)(127,163)(128,152)(129,141)(131,179)(132,168)(133,157)(134,146)(137,173)(138,162)(139,151)(142,178)(143,167)(144,156)(148,172)(149,161)(153,177)(154,166)(159,171)(164,176)(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), (1,165,31,135)(2,176,32,146)(3,127,33,157)(4,138,34,168)(5,149,35,179)(6,160,36,130)(7,171,37,141)(8,122,38,152)(9,133,39,163)(10,144,40,174)(11,155,41,125)(12,166,42,136)(13,177,43,147)(14,128,44,158)(15,139,45,169)(16,150,46,180)(17,161,47,131)(18,172,48,142)(19,123,49,153)(20,134,50,164)(21,145,51,175)(22,156,52,126)(23,167,53,137)(24,178,54,148)(25,129,55,159)(26,140,56,170)(27,151,57,121)(28,162,58,132)(29,173,59,143)(30,124,60,154)(61,182,91,212)(62,193,92,223)(63,204,93,234)(64,215,94,185)(65,226,95,196)(66,237,96,207)(67,188,97,218)(68,199,98,229)(69,210,99,240)(70,221,100,191)(71,232,101,202)(72,183,102,213)(73,194,103,224)(74,205,104,235)(75,216,105,186)(76,227,106,197)(77,238,107,208)(78,189,108,219)(79,200,109,230)(80,211,110,181)(81,222,111,192)(82,233,112,203)(83,184,113,214)(84,195,114,225)(85,206,115,236)(86,217,116,187)(87,228,117,198)(88,239,118,209)(89,190,119,220)(90,201,120,231), (1,102)(2,73)(3,104)(4,75)(5,106)(6,77)(7,108)(8,79)(9,110)(10,81)(11,112)(12,83)(13,114)(14,85)(15,116)(16,87)(17,118)(18,89)(19,120)(20,91)(21,62)(22,93)(23,64)(24,95)(25,66)(26,97)(27,68)(28,99)(29,70)(30,101)(31,72)(32,103)(33,74)(34,105)(35,76)(36,107)(37,78)(38,109)(39,80)(40,111)(41,82)(42,113)(43,84)(44,115)(45,86)(46,117)(47,88)(48,119)(49,90)(50,61)(51,92)(52,63)(53,94)(54,65)(55,96)(56,67)(57,98)(58,69)(59,100)(60,71)(121,184)(122,215)(123,186)(124,217)(125,188)(126,219)(127,190)(128,221)(129,192)(130,223)(131,194)(132,225)(133,196)(134,227)(135,198)(136,229)(137,200)(138,231)(139,202)(140,233)(141,204)(142,235)(143,206)(144,237)(145,208)(146,239)(147,210)(148,181)(149,212)(150,183)(151,214)(152,185)(153,216)(154,187)(155,218)(156,189)(157,220)(158,191)(159,222)(160,193)(161,224)(162,195)(163,226)(164,197)(165,228)(166,199)(167,230)(168,201)(169,232)(170,203)(171,234)(172,205)(173,236)(174,207)(175,238)(176,209)(177,240)(178,211)(179,182)(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)(61,103)(62,92)(63,81)(64,70)(65,119)(66,108)(67,97)(68,86)(69,75)(71,113)(72,102)(73,91)(74,80)(76,118)(77,107)(78,96)(79,85)(82,112)(83,101)(84,90)(87,117)(88,106)(89,95)(93,111)(94,100)(98,116)(99,105)(104,110)(109,115)(114,120)(121,169)(122,158)(123,147)(124,136)(126,174)(127,163)(128,152)(129,141)(131,179)(132,168)(133,157)(134,146)(137,173)(138,162)(139,151)(142,178)(143,167)(144,156)(148,172)(149,161)(153,177)(154,166)(159,171)(164,176)(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), (1,165,31,135)(2,176,32,146)(3,127,33,157)(4,138,34,168)(5,149,35,179)(6,160,36,130)(7,171,37,141)(8,122,38,152)(9,133,39,163)(10,144,40,174)(11,155,41,125)(12,166,42,136)(13,177,43,147)(14,128,44,158)(15,139,45,169)(16,150,46,180)(17,161,47,131)(18,172,48,142)(19,123,49,153)(20,134,50,164)(21,145,51,175)(22,156,52,126)(23,167,53,137)(24,178,54,148)(25,129,55,159)(26,140,56,170)(27,151,57,121)(28,162,58,132)(29,173,59,143)(30,124,60,154)(61,182,91,212)(62,193,92,223)(63,204,93,234)(64,215,94,185)(65,226,95,196)(66,237,96,207)(67,188,97,218)(68,199,98,229)(69,210,99,240)(70,221,100,191)(71,232,101,202)(72,183,102,213)(73,194,103,224)(74,205,104,235)(75,216,105,186)(76,227,106,197)(77,238,107,208)(78,189,108,219)(79,200,109,230)(80,211,110,181)(81,222,111,192)(82,233,112,203)(83,184,113,214)(84,195,114,225)(85,206,115,236)(86,217,116,187)(87,228,117,198)(88,239,118,209)(89,190,119,220)(90,201,120,231), (1,102)(2,73)(3,104)(4,75)(5,106)(6,77)(7,108)(8,79)(9,110)(10,81)(11,112)(12,83)(13,114)(14,85)(15,116)(16,87)(17,118)(18,89)(19,120)(20,91)(21,62)(22,93)(23,64)(24,95)(25,66)(26,97)(27,68)(28,99)(29,70)(30,101)(31,72)(32,103)(33,74)(34,105)(35,76)(36,107)(37,78)(38,109)(39,80)(40,111)(41,82)(42,113)(43,84)(44,115)(45,86)(46,117)(47,88)(48,119)(49,90)(50,61)(51,92)(52,63)(53,94)(54,65)(55,96)(56,67)(57,98)(58,69)(59,100)(60,71)(121,184)(122,215)(123,186)(124,217)(125,188)(126,219)(127,190)(128,221)(129,192)(130,223)(131,194)(132,225)(133,196)(134,227)(135,198)(136,229)(137,200)(138,231)(139,202)(140,233)(141,204)(142,235)(143,206)(144,237)(145,208)(146,239)(147,210)(148,181)(149,212)(150,183)(151,214)(152,185)(153,216)(154,187)(155,218)(156,189)(157,220)(158,191)(159,222)(160,193)(161,224)(162,195)(163,226)(164,197)(165,228)(166,199)(167,230)(168,201)(169,232)(170,203)(171,234)(172,205)(173,236)(174,207)(175,238)(176,209)(177,240)(178,211)(179,182)(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),(61,103),(62,92),(63,81),(64,70),(65,119),(66,108),(67,97),(68,86),(69,75),(71,113),(72,102),(73,91),(74,80),(76,118),(77,107),(78,96),(79,85),(82,112),(83,101),(84,90),(87,117),(88,106),(89,95),(93,111),(94,100),(98,116),(99,105),(104,110),(109,115),(114,120),(121,169),(122,158),(123,147),(124,136),(126,174),(127,163),(128,152),(129,141),(131,179),(132,168),(133,157),(134,146),(137,173),(138,162),(139,151),(142,178),(143,167),(144,156),(148,172),(149,161),(153,177),(154,166),(159,171),(164,176),(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)], [(1,165,31,135),(2,176,32,146),(3,127,33,157),(4,138,34,168),(5,149,35,179),(6,160,36,130),(7,171,37,141),(8,122,38,152),(9,133,39,163),(10,144,40,174),(11,155,41,125),(12,166,42,136),(13,177,43,147),(14,128,44,158),(15,139,45,169),(16,150,46,180),(17,161,47,131),(18,172,48,142),(19,123,49,153),(20,134,50,164),(21,145,51,175),(22,156,52,126),(23,167,53,137),(24,178,54,148),(25,129,55,159),(26,140,56,170),(27,151,57,121),(28,162,58,132),(29,173,59,143),(30,124,60,154),(61,182,91,212),(62,193,92,223),(63,204,93,234),(64,215,94,185),(65,226,95,196),(66,237,96,207),(67,188,97,218),(68,199,98,229),(69,210,99,240),(70,221,100,191),(71,232,101,202),(72,183,102,213),(73,194,103,224),(74,205,104,235),(75,216,105,186),(76,227,106,197),(77,238,107,208),(78,189,108,219),(79,200,109,230),(80,211,110,181),(81,222,111,192),(82,233,112,203),(83,184,113,214),(84,195,114,225),(85,206,115,236),(86,217,116,187),(87,228,117,198),(88,239,118,209),(89,190,119,220),(90,201,120,231)], [(1,102),(2,73),(3,104),(4,75),(5,106),(6,77),(7,108),(8,79),(9,110),(10,81),(11,112),(12,83),(13,114),(14,85),(15,116),(16,87),(17,118),(18,89),(19,120),(20,91),(21,62),(22,93),(23,64),(24,95),(25,66),(26,97),(27,68),(28,99),(29,70),(30,101),(31,72),(32,103),(33,74),(34,105),(35,76),(36,107),(37,78),(38,109),(39,80),(40,111),(41,82),(42,113),(43,84),(44,115),(45,86),(46,117),(47,88),(48,119),(49,90),(50,61),(51,92),(52,63),(53,94),(54,65),(55,96),(56,67),(57,98),(58,69),(59,100),(60,71),(121,184),(122,215),(123,186),(124,217),(125,188),(126,219),(127,190),(128,221),(129,192),(130,223),(131,194),(132,225),(133,196),(134,227),(135,198),(136,229),(137,200),(138,231),(139,202),(140,233),(141,204),(142,235),(143,206),(144,237),(145,208),(146,239),(147,210),(148,181),(149,212),(150,183),(151,214),(152,185),(153,216),(154,187),(155,218),(156,189),(157,220),(158,191),(159,222),(160,193),(161,224),(162,195),(163,226),(164,197),(165,228),(166,199),(167,230),(168,201),(169,232),(170,203),(171,234),(172,205),(173,236),(174,207),(175,238),(176,209),(177,240),(178,211),(179,182),(180,213)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 10 | 2 | 2 | 10 | 12 | 20 | 60 | 2 | 2 | 2 | 4 | 4 | 20 | 12 | 60 | 2 | 2 | 8 | 8 | 4 | 10 | 10 | 20 | 20 | 4 | 4 | 4 | 4 | 24 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | Q8.14D6 | C2×S3×D5 | SD16⋊D5 | D5×C3⋊D4 | C60.8C23 |
| kernel | C60.8C23 | C20.32D6 | C15⋊Q16 | C3⋊Dic20 | C5×D4.S3 | D4.D15 | D5×Dic6 | C3×D4⋊2D5 | D4⋊2D5 | C3×Dic5 | C6×D5 | D4.S3 | Dic10 | C4×D5 | C5×D4 | C3⋊C8 | Dic6 | C3×D4 | Dic5 | D10 | C15 | D4 | 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 C60.8C23 ►in GL6(𝔽241)
| 52 | 52 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 99 | 0 | 0 |
| 0 | 0 | 142 | 198 | 0 | 0 |
| 0 | 0 | 118 | 234 | 142 | 142 |
| 0 | 0 | 94 | 79 | 99 | 43 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 151 | 228 | 240 | 0 |
| 0 | 0 | 128 | 68 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 81 | 220 | 0 | 0 |
| 0 | 0 | 60 | 160 | 0 | 0 |
| 0 | 0 | 35 | 135 | 5 | 19 |
| 0 | 0 | 189 | 84 | 24 | 236 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 225 | 153 | 5 | 0 |
| 0 | 0 | 162 | 71 | 0 | 5 |
| 0 | 0 | 101 | 4 | 16 | 88 |
| 0 | 0 | 146 | 108 | 79 | 170 |
G:=sub<GL(6,GF(241))| [52,189,0,0,0,0,52,240,0,0,0,0,0,0,99,142,118,94,0,0,99,198,234,79,0,0,0,0,142,99,0,0,0,0,142,43],[1,189,0,0,0,0,0,240,0,0,0,0,0,0,1,0,151,128,0,0,0,1,228,68,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,81,60,35,189,0,0,220,160,135,84,0,0,0,0,5,24,0,0,0,0,19,236],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,225,162,101,146,0,0,153,71,4,108,0,0,5,0,16,79,0,0,0,5,88,170] >;
C60.8C23 in GAP, Magma, Sage, TeX
C_{60}._8C_2^3 % in TeX
G:=Group("C60.8C2^3"); // GroupNames label
G:=SmallGroup(480,560);
// by ID
G=gap.SmallGroup(480,560);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,135,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^60=b^2=d^2=1,c^2=a^30,b*a*b=a^49,c*a*c^-1=a^11,d*a*d=a^31,b*c=c*b,d*b*d=a^30*b,d*c*d=a^45*c>;
// generators/relations