metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.10C23, Dic6.6D10, Dic10.23D6, Dic30.2C22, D4.D5⋊1S3, D4⋊2S3.D5, D4.2(S3×D5), C5⋊2C8.5D6, D4.D15⋊4C2, (C5×D4).19D6, (C4×S3).6D10, C15⋊Q16⋊2C2, C5⋊Dic12⋊1C2, (S3×C10).9D4, (C3×D4).2D10, C5⋊6(D4.D6), (S3×Dic10)⋊2C2, C30.172(C2×D4), C10.142(S3×D4), D6.7(C5⋊D4), D6.Dic5⋊2C2, C3⋊2(D4.9D10), C15⋊14(C8.C22), (S3×C20).4C22, C20.10(C22×S3), C15⋊3C8.2C22, (C5×Dic3).35D4, (D4×C15).4C22, C12.10(C22×D5), (C5×Dic6).2C22, Dic3.16(C5⋊D4), (C3×Dic10).2C22, C4.10(C2×S3×D5), (C3×D4.D5)⋊2C2, C2.23(S3×C5⋊D4), C6.45(C2×C5⋊D4), (C5×D4⋊2S3).1C2, (C3×C5⋊2C8).1C22, SmallGroup(480,562)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.10C23
G = < a,b,c,d | a60=c2=d2=1, b2=a30, bab-1=a19, cac=a41, dad=a31, bc=cb, dbd=a45b, dcd=a30c >
Subgroups: 540 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C30, C30, C8.C22, C5⋊2C8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C5×D4, C5×D4, C5×Q8, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C4.Dic5, D4.D5, D4.D5, C5⋊Q16, C2×Dic10, C5×C4○D4, D4.D6, C3×C5⋊2C8, C15⋊3C8, S3×Dic5, C15⋊Q8, C3×Dic10, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, Dic30, D4×C15, D4.9D10, D6.Dic5, C15⋊Q16, C5⋊Dic12, C3×D4.D5, D4.D15, S3×Dic10, C5×D4⋊2S3, C60.10C23
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D4.D6, C2×S3×D5, D4.9D10, S3×C5⋊D4, C60.10C23
►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)
(1 163 31 133)(2 122 32 152)(3 141 33 171)(4 160 34 130)(5 179 35 149)(6 138 36 168)(7 157 37 127)(8 176 38 146)(9 135 39 165)(10 154 40 124)(11 173 41 143)(12 132 42 162)(13 151 43 121)(14 170 44 140)(15 129 45 159)(16 148 46 178)(17 167 47 137)(18 126 48 156)(19 145 49 175)(20 164 50 134)(21 123 51 153)(22 142 52 172)(23 161 53 131)(24 180 54 150)(25 139 55 169)(26 158 56 128)(27 177 57 147)(28 136 58 166)(29 155 59 125)(30 174 60 144)(61 209 91 239)(62 228 92 198)(63 187 93 217)(64 206 94 236)(65 225 95 195)(66 184 96 214)(67 203 97 233)(68 222 98 192)(69 181 99 211)(70 200 100 230)(71 219 101 189)(72 238 102 208)(73 197 103 227)(74 216 104 186)(75 235 105 205)(76 194 106 224)(77 213 107 183)(78 232 108 202)(79 191 109 221)(80 210 110 240)(81 229 111 199)(82 188 112 218)(83 207 113 237)(84 226 114 196)(85 185 115 215)(86 204 116 234)(87 223 117 193)(88 182 118 212)(89 201 119 231)(90 220 120 190)
(2 42)(3 23)(5 45)(6 26)(8 48)(9 29)(11 51)(12 32)(14 54)(15 35)(17 57)(18 38)(20 60)(21 41)(24 44)(27 47)(30 50)(33 53)(36 56)(39 59)(61 71)(62 112)(63 93)(64 74)(65 115)(66 96)(67 77)(68 118)(69 99)(70 80)(72 102)(73 83)(75 105)(76 86)(78 108)(79 89)(81 111)(82 92)(84 114)(85 95)(87 117)(88 98)(90 120)(91 101)(94 104)(97 107)(100 110)(103 113)(106 116)(109 119)(122 162)(123 143)(125 165)(126 146)(128 168)(129 149)(131 171)(132 152)(134 174)(135 155)(137 177)(138 158)(140 180)(141 161)(144 164)(147 167)(150 170)(153 173)(156 176)(159 179)(181 211)(182 192)(183 233)(184 214)(185 195)(186 236)(187 217)(188 198)(189 239)(190 220)(191 201)(193 223)(194 204)(196 226)(197 207)(199 229)(200 210)(202 232)(203 213)(205 235)(206 216)(208 238)(209 219)(212 222)(215 225)(218 228)(221 231)(224 234)(227 237)(230 240)
(1 117)(2 88)(3 119)(4 90)(5 61)(6 92)(7 63)(8 94)(9 65)(10 96)(11 67)(12 98)(13 69)(14 100)(15 71)(16 102)(17 73)(18 104)(19 75)(20 106)(21 77)(22 108)(23 79)(24 110)(25 81)(26 112)(27 83)(28 114)(29 85)(30 116)(31 87)(32 118)(33 89)(34 120)(35 91)(36 62)(37 93)(38 64)(39 95)(40 66)(41 97)(42 68)(43 99)(44 70)(45 101)(46 72)(47 103)(48 74)(49 105)(50 76)(51 107)(52 78)(53 109)(54 80)(55 111)(56 82)(57 113)(58 84)(59 115)(60 86)(121 226)(122 197)(123 228)(124 199)(125 230)(126 201)(127 232)(128 203)(129 234)(130 205)(131 236)(132 207)(133 238)(134 209)(135 240)(136 211)(137 182)(138 213)(139 184)(140 215)(141 186)(142 217)(143 188)(144 219)(145 190)(146 221)(147 192)(148 223)(149 194)(150 225)(151 196)(152 227)(153 198)(154 229)(155 200)(156 231)(157 202)(158 233)(159 204)(160 235)(161 206)(162 237)(163 208)(164 239)(165 210)(166 181)(167 212)(168 183)(169 214)(170 185)(171 216)(172 187)(173 218)(174 189)(175 220)(176 191)(177 222)(178 193)(179 224)(180 195)
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), (1,163,31,133)(2,122,32,152)(3,141,33,171)(4,160,34,130)(5,179,35,149)(6,138,36,168)(7,157,37,127)(8,176,38,146)(9,135,39,165)(10,154,40,124)(11,173,41,143)(12,132,42,162)(13,151,43,121)(14,170,44,140)(15,129,45,159)(16,148,46,178)(17,167,47,137)(18,126,48,156)(19,145,49,175)(20,164,50,134)(21,123,51,153)(22,142,52,172)(23,161,53,131)(24,180,54,150)(25,139,55,169)(26,158,56,128)(27,177,57,147)(28,136,58,166)(29,155,59,125)(30,174,60,144)(61,209,91,239)(62,228,92,198)(63,187,93,217)(64,206,94,236)(65,225,95,195)(66,184,96,214)(67,203,97,233)(68,222,98,192)(69,181,99,211)(70,200,100,230)(71,219,101,189)(72,238,102,208)(73,197,103,227)(74,216,104,186)(75,235,105,205)(76,194,106,224)(77,213,107,183)(78,232,108,202)(79,191,109,221)(80,210,110,240)(81,229,111,199)(82,188,112,218)(83,207,113,237)(84,226,114,196)(85,185,115,215)(86,204,116,234)(87,223,117,193)(88,182,118,212)(89,201,119,231)(90,220,120,190), (2,42)(3,23)(5,45)(6,26)(8,48)(9,29)(11,51)(12,32)(14,54)(15,35)(17,57)(18,38)(20,60)(21,41)(24,44)(27,47)(30,50)(33,53)(36,56)(39,59)(61,71)(62,112)(63,93)(64,74)(65,115)(66,96)(67,77)(68,118)(69,99)(70,80)(72,102)(73,83)(75,105)(76,86)(78,108)(79,89)(81,111)(82,92)(84,114)(85,95)(87,117)(88,98)(90,120)(91,101)(94,104)(97,107)(100,110)(103,113)(106,116)(109,119)(122,162)(123,143)(125,165)(126,146)(128,168)(129,149)(131,171)(132,152)(134,174)(135,155)(137,177)(138,158)(140,180)(141,161)(144,164)(147,167)(150,170)(153,173)(156,176)(159,179)(181,211)(182,192)(183,233)(184,214)(185,195)(186,236)(187,217)(188,198)(189,239)(190,220)(191,201)(193,223)(194,204)(196,226)(197,207)(199,229)(200,210)(202,232)(203,213)(205,235)(206,216)(208,238)(209,219)(212,222)(215,225)(218,228)(221,231)(224,234)(227,237)(230,240), (1,117)(2,88)(3,119)(4,90)(5,61)(6,92)(7,63)(8,94)(9,65)(10,96)(11,67)(12,98)(13,69)(14,100)(15,71)(16,102)(17,73)(18,104)(19,75)(20,106)(21,77)(22,108)(23,79)(24,110)(25,81)(26,112)(27,83)(28,114)(29,85)(30,116)(31,87)(32,118)(33,89)(34,120)(35,91)(36,62)(37,93)(38,64)(39,95)(40,66)(41,97)(42,68)(43,99)(44,70)(45,101)(46,72)(47,103)(48,74)(49,105)(50,76)(51,107)(52,78)(53,109)(54,80)(55,111)(56,82)(57,113)(58,84)(59,115)(60,86)(121,226)(122,197)(123,228)(124,199)(125,230)(126,201)(127,232)(128,203)(129,234)(130,205)(131,236)(132,207)(133,238)(134,209)(135,240)(136,211)(137,182)(138,213)(139,184)(140,215)(141,186)(142,217)(143,188)(144,219)(145,190)(146,221)(147,192)(148,223)(149,194)(150,225)(151,196)(152,227)(153,198)(154,229)(155,200)(156,231)(157,202)(158,233)(159,204)(160,235)(161,206)(162,237)(163,208)(164,239)(165,210)(166,181)(167,212)(168,183)(169,214)(170,185)(171,216)(172,187)(173,218)(174,189)(175,220)(176,191)(177,222)(178,193)(179,224)(180,195)>;
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), (1,163,31,133)(2,122,32,152)(3,141,33,171)(4,160,34,130)(5,179,35,149)(6,138,36,168)(7,157,37,127)(8,176,38,146)(9,135,39,165)(10,154,40,124)(11,173,41,143)(12,132,42,162)(13,151,43,121)(14,170,44,140)(15,129,45,159)(16,148,46,178)(17,167,47,137)(18,126,48,156)(19,145,49,175)(20,164,50,134)(21,123,51,153)(22,142,52,172)(23,161,53,131)(24,180,54,150)(25,139,55,169)(26,158,56,128)(27,177,57,147)(28,136,58,166)(29,155,59,125)(30,174,60,144)(61,209,91,239)(62,228,92,198)(63,187,93,217)(64,206,94,236)(65,225,95,195)(66,184,96,214)(67,203,97,233)(68,222,98,192)(69,181,99,211)(70,200,100,230)(71,219,101,189)(72,238,102,208)(73,197,103,227)(74,216,104,186)(75,235,105,205)(76,194,106,224)(77,213,107,183)(78,232,108,202)(79,191,109,221)(80,210,110,240)(81,229,111,199)(82,188,112,218)(83,207,113,237)(84,226,114,196)(85,185,115,215)(86,204,116,234)(87,223,117,193)(88,182,118,212)(89,201,119,231)(90,220,120,190), (2,42)(3,23)(5,45)(6,26)(8,48)(9,29)(11,51)(12,32)(14,54)(15,35)(17,57)(18,38)(20,60)(21,41)(24,44)(27,47)(30,50)(33,53)(36,56)(39,59)(61,71)(62,112)(63,93)(64,74)(65,115)(66,96)(67,77)(68,118)(69,99)(70,80)(72,102)(73,83)(75,105)(76,86)(78,108)(79,89)(81,111)(82,92)(84,114)(85,95)(87,117)(88,98)(90,120)(91,101)(94,104)(97,107)(100,110)(103,113)(106,116)(109,119)(122,162)(123,143)(125,165)(126,146)(128,168)(129,149)(131,171)(132,152)(134,174)(135,155)(137,177)(138,158)(140,180)(141,161)(144,164)(147,167)(150,170)(153,173)(156,176)(159,179)(181,211)(182,192)(183,233)(184,214)(185,195)(186,236)(187,217)(188,198)(189,239)(190,220)(191,201)(193,223)(194,204)(196,226)(197,207)(199,229)(200,210)(202,232)(203,213)(205,235)(206,216)(208,238)(209,219)(212,222)(215,225)(218,228)(221,231)(224,234)(227,237)(230,240), (1,117)(2,88)(3,119)(4,90)(5,61)(6,92)(7,63)(8,94)(9,65)(10,96)(11,67)(12,98)(13,69)(14,100)(15,71)(16,102)(17,73)(18,104)(19,75)(20,106)(21,77)(22,108)(23,79)(24,110)(25,81)(26,112)(27,83)(28,114)(29,85)(30,116)(31,87)(32,118)(33,89)(34,120)(35,91)(36,62)(37,93)(38,64)(39,95)(40,66)(41,97)(42,68)(43,99)(44,70)(45,101)(46,72)(47,103)(48,74)(49,105)(50,76)(51,107)(52,78)(53,109)(54,80)(55,111)(56,82)(57,113)(58,84)(59,115)(60,86)(121,226)(122,197)(123,228)(124,199)(125,230)(126,201)(127,232)(128,203)(129,234)(130,205)(131,236)(132,207)(133,238)(134,209)(135,240)(136,211)(137,182)(138,213)(139,184)(140,215)(141,186)(142,217)(143,188)(144,219)(145,190)(146,221)(147,192)(148,223)(149,194)(150,225)(151,196)(152,227)(153,198)(154,229)(155,200)(156,231)(157,202)(158,233)(159,204)(160,235)(161,206)(162,237)(163,208)(164,239)(165,210)(166,181)(167,212)(168,183)(169,214)(170,185)(171,216)(172,187)(173,218)(174,189)(175,220)(176,191)(177,222)(178,193)(179,224)(180,195) );
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)], [(1,163,31,133),(2,122,32,152),(3,141,33,171),(4,160,34,130),(5,179,35,149),(6,138,36,168),(7,157,37,127),(8,176,38,146),(9,135,39,165),(10,154,40,124),(11,173,41,143),(12,132,42,162),(13,151,43,121),(14,170,44,140),(15,129,45,159),(16,148,46,178),(17,167,47,137),(18,126,48,156),(19,145,49,175),(20,164,50,134),(21,123,51,153),(22,142,52,172),(23,161,53,131),(24,180,54,150),(25,139,55,169),(26,158,56,128),(27,177,57,147),(28,136,58,166),(29,155,59,125),(30,174,60,144),(61,209,91,239),(62,228,92,198),(63,187,93,217),(64,206,94,236),(65,225,95,195),(66,184,96,214),(67,203,97,233),(68,222,98,192),(69,181,99,211),(70,200,100,230),(71,219,101,189),(72,238,102,208),(73,197,103,227),(74,216,104,186),(75,235,105,205),(76,194,106,224),(77,213,107,183),(78,232,108,202),(79,191,109,221),(80,210,110,240),(81,229,111,199),(82,188,112,218),(83,207,113,237),(84,226,114,196),(85,185,115,215),(86,204,116,234),(87,223,117,193),(88,182,118,212),(89,201,119,231),(90,220,120,190)], [(2,42),(3,23),(5,45),(6,26),(8,48),(9,29),(11,51),(12,32),(14,54),(15,35),(17,57),(18,38),(20,60),(21,41),(24,44),(27,47),(30,50),(33,53),(36,56),(39,59),(61,71),(62,112),(63,93),(64,74),(65,115),(66,96),(67,77),(68,118),(69,99),(70,80),(72,102),(73,83),(75,105),(76,86),(78,108),(79,89),(81,111),(82,92),(84,114),(85,95),(87,117),(88,98),(90,120),(91,101),(94,104),(97,107),(100,110),(103,113),(106,116),(109,119),(122,162),(123,143),(125,165),(126,146),(128,168),(129,149),(131,171),(132,152),(134,174),(135,155),(137,177),(138,158),(140,180),(141,161),(144,164),(147,167),(150,170),(153,173),(156,176),(159,179),(181,211),(182,192),(183,233),(184,214),(185,195),(186,236),(187,217),(188,198),(189,239),(190,220),(191,201),(193,223),(194,204),(196,226),(197,207),(199,229),(200,210),(202,232),(203,213),(205,235),(206,216),(208,238),(209,219),(212,222),(215,225),(218,228),(221,231),(224,234),(227,237),(230,240)], [(1,117),(2,88),(3,119),(4,90),(5,61),(6,92),(7,63),(8,94),(9,65),(10,96),(11,67),(12,98),(13,69),(14,100),(15,71),(16,102),(17,73),(18,104),(19,75),(20,106),(21,77),(22,108),(23,79),(24,110),(25,81),(26,112),(27,83),(28,114),(29,85),(30,116),(31,87),(32,118),(33,89),(34,120),(35,91),(36,62),(37,93),(38,64),(39,95),(40,66),(41,97),(42,68),(43,99),(44,70),(45,101),(46,72),(47,103),(48,74),(49,105),(50,76),(51,107),(52,78),(53,109),(54,80),(55,111),(56,82),(57,113),(58,84),(59,115),(60,86),(121,226),(122,197),(123,228),(124,199),(125,230),(126,201),(127,232),(128,203),(129,234),(130,205),(131,236),(132,207),(133,238),(134,209),(135,240),(136,211),(137,182),(138,213),(139,184),(140,215),(141,186),(142,217),(143,188),(144,219),(145,190),(146,221),(147,192),(148,223),(149,194),(150,225),(151,196),(152,227),(153,198),(154,229),(155,200),(156,231),(157,202),(158,233),(159,204),(160,235),(161,206),(162,237),(163,208),(164,239),(165,210),(166,181),(167,212),(168,183),(169,214),(170,185),(171,216),(172,187),(173,218),(174,189),(175,220),(176,191),(177,222),(178,193),(179,224),(180,195)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 4 | 6 | 2 | 2 | 6 | 12 | 20 | 60 | 2 | 2 | 2 | 8 | 20 | 60 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 40 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
48 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 | C5⋊D4 | C5⋊D4 | C8.C22 | S3×D4 | S3×D5 | D4.D6 | C2×S3×D5 | D4.9D10 | S3×C5⋊D4 | C60.10C23 |
| kernel | C60.10C23 | D6.Dic5 | C15⋊Q16 | C5⋊Dic12 | C3×D4.D5 | D4.D15 | S3×Dic10 | C5×D4⋊2S3 | D4.D5 | C5×Dic3 | S3×C10 | D4⋊2S3 | C5⋊2C8 | Dic10 | C5×D4 | Dic6 | C4×S3 | C3×D4 | Dic3 | D6 | C15 | C10 | D4 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of C60.10C23 ►in GL8(𝔽241)
| 2 | 137 | 192 | 103 | 0 | 0 | 0 | 0 |
| 104 | 137 | 0 | 54 | 0 | 0 | 0 | 0 |
| 177 | 64 | 0 | 52 | 0 | 0 | 0 | 0 |
| 67 | 0 | 190 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 160 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 239 | 1 | 238 |
| 0 | 0 | 0 | 0 | 0 | 160 | 81 | 240 |
| 220 | 166 | 0 | 0 | 0 | 0 | 0 | 0 |
| 38 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 59 | 203 | 0 | 0 | 0 | 0 |
| 0 | 0 | 155 | 182 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 191 | 0 | 0 |
| 0 | 0 | 0 | 0 | 82 | 219 | 0 | 0 |
| 0 | 0 | 0 | 0 | 225 | 113 | 219 | 82 |
| 0 | 0 | 0 | 0 | 128 | 45 | 191 | 22 |
| 240 | 0 | 103 | 138 | 0 | 0 | 0 | 0 |
| 0 | 240 | 103 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 2 | 160 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 160 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 239 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(8,GF(241))| [2,104,177,67,0,0,0,0,137,137,64,0,0,0,0,0,192,0,0,190,0,0,0,0,103,54,52,51,0,0,0,0,0,0,0,0,1,3,3,0,0,0,0,0,160,240,239,160,0,0,0,0,0,0,1,81,0,0,0,0,0,0,238,240],[220,38,0,0,0,0,0,0,166,21,0,0,0,0,0,0,0,0,59,155,0,0,0,0,0,0,203,182,0,0,0,0,0,0,0,0,22,82,225,128,0,0,0,0,191,219,113,45,0,0,0,0,0,0,219,191,0,0,0,0,0,0,82,22],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,103,103,1,0,0,0,0,0,138,0,0,1,0,0,0,0,0,0,0,0,1,0,3,2,0,0,0,0,0,1,0,160,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,160,239,240,0,0,0,0,0,0,3,0,240] >;
C60.10C23 in GAP, Magma, Sage, TeX
C_{60}._{10}C_2^3 % in TeX
G:=Group("C60.10C2^3"); // GroupNames label
G:=SmallGroup(480,562);
// by ID
G=gap.SmallGroup(480,562);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^60=c^2=d^2=1,b^2=a^30,b*a*b^-1=a^19,c*a*c=a^41,d*a*d=a^31,b*c=c*b,d*b*d=a^45*b,d*c*d=a^30*c>;
// generators/relations