metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.16C23, Dic10.9D6, D60.4C22, Dic6.24D10, D4⋊D15⋊6C2, D4.S3⋊7D5, C3⋊C8.14D10, (C5×D4).8D6, D4⋊2D5⋊3S3, C15⋊14(C4○D8), (C6×D5).11D4, D4.12(S3×D5), (C4×D5).46D6, C15⋊Q16⋊3C2, C6.145(D4×D5), C12.28D10⋊2C2, (C3×D4).23D10, C30.178(C2×D4), C15⋊SD16⋊3C2, C5⋊3(Q8.13D6), C3⋊6(SD16⋊3D5), C20.16(C22×S3), C15⋊3C8.4C22, (C3×Dic5).69D4, (D5×C12).8C22, C12.16(C22×D5), D10.10(C3⋊D4), (D4×C15).10C22, (C5×Dic6).4C22, Dic5.41(C3⋊D4), (C3×Dic10).5C22, (D5×C3⋊C8)⋊4C2, C4.16(C2×S3×D5), (C5×D4.S3)⋊6C2, C2.27(D5×C3⋊D4), (C5×C3⋊C8).4C22, (C3×D4⋊2D5)⋊3C2, C10.48(C2×C3⋊D4), SmallGroup(480,568)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.16C23
G = < a,b,c,d | a60=b2=d2=1, c2=a30, bab=a49, cac-1=a11, dad=a31, cbc-1=dbd=a30b, dcd=a45c >
Subgroups: 668 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, D15, C30, C30, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5, C60, C6×D5, D30, C2×C30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8⋊2D5, Q8.13D6, C5×C3⋊C8, C15⋊3C8, D30.C2, C3⋊D20, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×Dic6, D60, D4×C15, SD16⋊3D5, D5×C3⋊C8, C15⋊SD16, C15⋊Q16, C5×D4.S3, D4⋊D15, C12.28D10, C3×D4⋊2D5, C60.16C23
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4○D8, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.13D6, C2×S3×D5, SD16⋊3D5, D5×C3⋊D4, C60.16C23
►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 91)(62 80)(63 69)(64 118)(65 107)(66 96)(67 85)(68 74)(70 112)(71 101)(72 90)(73 79)(75 117)(76 106)(77 95)(78 84)(81 111)(82 100)(83 89)(86 116)(87 105)(88 94)(92 110)(93 99)(97 115)(98 104)(102 120)(103 109)(108 114)(113 119)(121 175)(122 164)(123 153)(124 142)(125 131)(126 180)(127 169)(128 158)(129 147)(130 136)(132 174)(133 163)(134 152)(135 141)(137 179)(138 168)(139 157)(140 146)(143 173)(144 162)(145 151)(148 178)(149 167)(150 156)(154 172)(155 161)(159 177)(160 166)(165 171)(170 176)(181 217)(182 206)(183 195)(185 233)(186 222)(187 211)(188 200)(190 238)(191 227)(192 216)(193 205)(196 232)(197 221)(198 210)(201 237)(202 226)(203 215)(207 231)(208 220)(212 236)(213 225)(218 230)(223 235)(228 240)
(1 173 31 143)(2 124 32 154)(3 135 33 165)(4 146 34 176)(5 157 35 127)(6 168 36 138)(7 179 37 149)(8 130 38 160)(9 141 39 171)(10 152 40 122)(11 163 41 133)(12 174 42 144)(13 125 43 155)(14 136 44 166)(15 147 45 177)(16 158 46 128)(17 169 47 139)(18 180 48 150)(19 131 49 161)(20 142 50 172)(21 153 51 123)(22 164 52 134)(23 175 53 145)(24 126 54 156)(25 137 55 167)(26 148 56 178)(27 159 57 129)(28 170 58 140)(29 121 59 151)(30 132 60 162)(61 204 91 234)(62 215 92 185)(63 226 93 196)(64 237 94 207)(65 188 95 218)(66 199 96 229)(67 210 97 240)(68 221 98 191)(69 232 99 202)(70 183 100 213)(71 194 101 224)(72 205 102 235)(73 216 103 186)(74 227 104 197)(75 238 105 208)(76 189 106 219)(77 200 107 230)(78 211 108 181)(79 222 109 192)(80 233 110 203)(81 184 111 214)(82 195 112 225)(83 206 113 236)(84 217 114 187)(85 228 115 198)(86 239 116 209)(87 190 117 220)(88 201 118 231)(89 212 119 182)(90 223 120 193)
(1 76)(2 107)(3 78)(4 109)(5 80)(6 111)(7 82)(8 113)(9 84)(10 115)(11 86)(12 117)(13 88)(14 119)(15 90)(16 61)(17 92)(18 63)(19 94)(20 65)(21 96)(22 67)(23 98)(24 69)(25 100)(26 71)(27 102)(28 73)(29 104)(30 75)(31 106)(32 77)(33 108)(34 79)(35 110)(36 81)(37 112)(38 83)(39 114)(40 85)(41 116)(42 87)(43 118)(44 89)(45 120)(46 91)(47 62)(48 93)(49 64)(50 95)(51 66)(52 97)(53 68)(54 99)(55 70)(56 101)(57 72)(58 103)(59 74)(60 105)(121 212)(122 183)(123 214)(124 185)(125 216)(126 187)(127 218)(128 189)(129 220)(130 191)(131 222)(132 193)(133 224)(134 195)(135 226)(136 197)(137 228)(138 199)(139 230)(140 201)(141 232)(142 203)(143 234)(144 205)(145 236)(146 207)(147 238)(148 209)(149 240)(150 211)(151 182)(152 213)(153 184)(154 215)(155 186)(156 217)(157 188)(158 219)(159 190)(160 221)(161 192)(162 223)(163 194)(164 225)(165 196)(166 227)(167 198)(168 229)(169 200)(170 231)(171 202)(172 233)(173 204)(174 235)(175 206)(176 237)(177 208)(178 239)(179 210)(180 181)
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)(61,91)(62,80)(63,69)(64,118)(65,107)(66,96)(67,85)(68,74)(70,112)(71,101)(72,90)(73,79)(75,117)(76,106)(77,95)(78,84)(81,111)(82,100)(83,89)(86,116)(87,105)(88,94)(92,110)(93,99)(97,115)(98,104)(102,120)(103,109)(108,114)(113,119)(121,175)(122,164)(123,153)(124,142)(125,131)(126,180)(127,169)(128,158)(129,147)(130,136)(132,174)(133,163)(134,152)(135,141)(137,179)(138,168)(139,157)(140,146)(143,173)(144,162)(145,151)(148,178)(149,167)(150,156)(154,172)(155,161)(159,177)(160,166)(165,171)(170,176)(181,217)(182,206)(183,195)(185,233)(186,222)(187,211)(188,200)(190,238)(191,227)(192,216)(193,205)(196,232)(197,221)(198,210)(201,237)(202,226)(203,215)(207,231)(208,220)(212,236)(213,225)(218,230)(223,235)(228,240), (1,173,31,143)(2,124,32,154)(3,135,33,165)(4,146,34,176)(5,157,35,127)(6,168,36,138)(7,179,37,149)(8,130,38,160)(9,141,39,171)(10,152,40,122)(11,163,41,133)(12,174,42,144)(13,125,43,155)(14,136,44,166)(15,147,45,177)(16,158,46,128)(17,169,47,139)(18,180,48,150)(19,131,49,161)(20,142,50,172)(21,153,51,123)(22,164,52,134)(23,175,53,145)(24,126,54,156)(25,137,55,167)(26,148,56,178)(27,159,57,129)(28,170,58,140)(29,121,59,151)(30,132,60,162)(61,204,91,234)(62,215,92,185)(63,226,93,196)(64,237,94,207)(65,188,95,218)(66,199,96,229)(67,210,97,240)(68,221,98,191)(69,232,99,202)(70,183,100,213)(71,194,101,224)(72,205,102,235)(73,216,103,186)(74,227,104,197)(75,238,105,208)(76,189,106,219)(77,200,107,230)(78,211,108,181)(79,222,109,192)(80,233,110,203)(81,184,111,214)(82,195,112,225)(83,206,113,236)(84,217,114,187)(85,228,115,198)(86,239,116,209)(87,190,117,220)(88,201,118,231)(89,212,119,182)(90,223,120,193), (1,76)(2,107)(3,78)(4,109)(5,80)(6,111)(7,82)(8,113)(9,84)(10,115)(11,86)(12,117)(13,88)(14,119)(15,90)(16,61)(17,92)(18,63)(19,94)(20,65)(21,96)(22,67)(23,98)(24,69)(25,100)(26,71)(27,102)(28,73)(29,104)(30,75)(31,106)(32,77)(33,108)(34,79)(35,110)(36,81)(37,112)(38,83)(39,114)(40,85)(41,116)(42,87)(43,118)(44,89)(45,120)(46,91)(47,62)(48,93)(49,64)(50,95)(51,66)(52,97)(53,68)(54,99)(55,70)(56,101)(57,72)(58,103)(59,74)(60,105)(121,212)(122,183)(123,214)(124,185)(125,216)(126,187)(127,218)(128,189)(129,220)(130,191)(131,222)(132,193)(133,224)(134,195)(135,226)(136,197)(137,228)(138,199)(139,230)(140,201)(141,232)(142,203)(143,234)(144,205)(145,236)(146,207)(147,238)(148,209)(149,240)(150,211)(151,182)(152,213)(153,184)(154,215)(155,186)(156,217)(157,188)(158,219)(159,190)(160,221)(161,192)(162,223)(163,194)(164,225)(165,196)(166,227)(167,198)(168,229)(169,200)(170,231)(171,202)(172,233)(173,204)(174,235)(175,206)(176,237)(177,208)(178,239)(179,210)(180,181)>;
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)(61,91)(62,80)(63,69)(64,118)(65,107)(66,96)(67,85)(68,74)(70,112)(71,101)(72,90)(73,79)(75,117)(76,106)(77,95)(78,84)(81,111)(82,100)(83,89)(86,116)(87,105)(88,94)(92,110)(93,99)(97,115)(98,104)(102,120)(103,109)(108,114)(113,119)(121,175)(122,164)(123,153)(124,142)(125,131)(126,180)(127,169)(128,158)(129,147)(130,136)(132,174)(133,163)(134,152)(135,141)(137,179)(138,168)(139,157)(140,146)(143,173)(144,162)(145,151)(148,178)(149,167)(150,156)(154,172)(155,161)(159,177)(160,166)(165,171)(170,176)(181,217)(182,206)(183,195)(185,233)(186,222)(187,211)(188,200)(190,238)(191,227)(192,216)(193,205)(196,232)(197,221)(198,210)(201,237)(202,226)(203,215)(207,231)(208,220)(212,236)(213,225)(218,230)(223,235)(228,240), (1,173,31,143)(2,124,32,154)(3,135,33,165)(4,146,34,176)(5,157,35,127)(6,168,36,138)(7,179,37,149)(8,130,38,160)(9,141,39,171)(10,152,40,122)(11,163,41,133)(12,174,42,144)(13,125,43,155)(14,136,44,166)(15,147,45,177)(16,158,46,128)(17,169,47,139)(18,180,48,150)(19,131,49,161)(20,142,50,172)(21,153,51,123)(22,164,52,134)(23,175,53,145)(24,126,54,156)(25,137,55,167)(26,148,56,178)(27,159,57,129)(28,170,58,140)(29,121,59,151)(30,132,60,162)(61,204,91,234)(62,215,92,185)(63,226,93,196)(64,237,94,207)(65,188,95,218)(66,199,96,229)(67,210,97,240)(68,221,98,191)(69,232,99,202)(70,183,100,213)(71,194,101,224)(72,205,102,235)(73,216,103,186)(74,227,104,197)(75,238,105,208)(76,189,106,219)(77,200,107,230)(78,211,108,181)(79,222,109,192)(80,233,110,203)(81,184,111,214)(82,195,112,225)(83,206,113,236)(84,217,114,187)(85,228,115,198)(86,239,116,209)(87,190,117,220)(88,201,118,231)(89,212,119,182)(90,223,120,193), (1,76)(2,107)(3,78)(4,109)(5,80)(6,111)(7,82)(8,113)(9,84)(10,115)(11,86)(12,117)(13,88)(14,119)(15,90)(16,61)(17,92)(18,63)(19,94)(20,65)(21,96)(22,67)(23,98)(24,69)(25,100)(26,71)(27,102)(28,73)(29,104)(30,75)(31,106)(32,77)(33,108)(34,79)(35,110)(36,81)(37,112)(38,83)(39,114)(40,85)(41,116)(42,87)(43,118)(44,89)(45,120)(46,91)(47,62)(48,93)(49,64)(50,95)(51,66)(52,97)(53,68)(54,99)(55,70)(56,101)(57,72)(58,103)(59,74)(60,105)(121,212)(122,183)(123,214)(124,185)(125,216)(126,187)(127,218)(128,189)(129,220)(130,191)(131,222)(132,193)(133,224)(134,195)(135,226)(136,197)(137,228)(138,199)(139,230)(140,201)(141,232)(142,203)(143,234)(144,205)(145,236)(146,207)(147,238)(148,209)(149,240)(150,211)(151,182)(152,213)(153,184)(154,215)(155,186)(156,217)(157,188)(158,219)(159,190)(160,221)(161,192)(162,223)(163,194)(164,225)(165,196)(166,227)(167,198)(168,229)(169,200)(170,231)(171,202)(172,233)(173,204)(174,235)(175,206)(176,237)(177,208)(178,239)(179,210)(180,181) );
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),(61,91),(62,80),(63,69),(64,118),(65,107),(66,96),(67,85),(68,74),(70,112),(71,101),(72,90),(73,79),(75,117),(76,106),(77,95),(78,84),(81,111),(82,100),(83,89),(86,116),(87,105),(88,94),(92,110),(93,99),(97,115),(98,104),(102,120),(103,109),(108,114),(113,119),(121,175),(122,164),(123,153),(124,142),(125,131),(126,180),(127,169),(128,158),(129,147),(130,136),(132,174),(133,163),(134,152),(135,141),(137,179),(138,168),(139,157),(140,146),(143,173),(144,162),(145,151),(148,178),(149,167),(150,156),(154,172),(155,161),(159,177),(160,166),(165,171),(170,176),(181,217),(182,206),(183,195),(185,233),(186,222),(187,211),(188,200),(190,238),(191,227),(192,216),(193,205),(196,232),(197,221),(198,210),(201,237),(202,226),(203,215),(207,231),(208,220),(212,236),(213,225),(218,230),(223,235),(228,240)], [(1,173,31,143),(2,124,32,154),(3,135,33,165),(4,146,34,176),(5,157,35,127),(6,168,36,138),(7,179,37,149),(8,130,38,160),(9,141,39,171),(10,152,40,122),(11,163,41,133),(12,174,42,144),(13,125,43,155),(14,136,44,166),(15,147,45,177),(16,158,46,128),(17,169,47,139),(18,180,48,150),(19,131,49,161),(20,142,50,172),(21,153,51,123),(22,164,52,134),(23,175,53,145),(24,126,54,156),(25,137,55,167),(26,148,56,178),(27,159,57,129),(28,170,58,140),(29,121,59,151),(30,132,60,162),(61,204,91,234),(62,215,92,185),(63,226,93,196),(64,237,94,207),(65,188,95,218),(66,199,96,229),(67,210,97,240),(68,221,98,191),(69,232,99,202),(70,183,100,213),(71,194,101,224),(72,205,102,235),(73,216,103,186),(74,227,104,197),(75,238,105,208),(76,189,106,219),(77,200,107,230),(78,211,108,181),(79,222,109,192),(80,233,110,203),(81,184,111,214),(82,195,112,225),(83,206,113,236),(84,217,114,187),(85,228,115,198),(86,239,116,209),(87,190,117,220),(88,201,118,231),(89,212,119,182),(90,223,120,193)], [(1,76),(2,107),(3,78),(4,109),(5,80),(6,111),(7,82),(8,113),(9,84),(10,115),(11,86),(12,117),(13,88),(14,119),(15,90),(16,61),(17,92),(18,63),(19,94),(20,65),(21,96),(22,67),(23,98),(24,69),(25,100),(26,71),(27,102),(28,73),(29,104),(30,75),(31,106),(32,77),(33,108),(34,79),(35,110),(36,81),(37,112),(38,83),(39,114),(40,85),(41,116),(42,87),(43,118),(44,89),(45,120),(46,91),(47,62),(48,93),(49,64),(50,95),(51,66),(52,97),(53,68),(54,99),(55,70),(56,101),(57,72),(58,103),(59,74),(60,105),(121,212),(122,183),(123,214),(124,185),(125,216),(126,187),(127,218),(128,189),(129,220),(130,191),(131,222),(132,193),(133,224),(134,195),(135,226),(136,197),(137,228),(138,199),(139,230),(140,201),(141,232),(142,203),(143,234),(144,205),(145,236),(146,207),(147,238),(148,209),(149,240),(150,211),(151,182),(152,213),(153,184),(154,215),(155,186),(156,217),(157,188),(158,219),(159,190),(160,221),(161,192),(162,223),(163,194),(164,225),(165,196),(166,227),(167,198),(168,229),(169,200),(170,231),(171,202),(172,233),(173,204),(174,235),(175,206),(176,237),(177,208),(178,239),(179,210),(180,181)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 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 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 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 | 60 | 2 | 2 | 5 | 5 | 12 | 20 | 2 | 2 | 2 | 4 | 4 | 20 | 6 | 6 | 30 | 30 | 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 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | C4○D8 | S3×D5 | D4×D5 | Q8.13D6 | C2×S3×D5 | SD16⋊3D5 | D5×C3⋊D4 | C60.16C23 |
| kernel | C60.16C23 | D5×C3⋊C8 | C15⋊SD16 | C15⋊Q16 | C5×D4.S3 | D4⋊D15 | C12.28D10 | 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 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of C60.16C23 ►in GL6(𝔽241)
| 190 | 190 | 0 | 0 | 0 | 0 |
| 51 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 101 | 177 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 51 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 43 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 73 | 55 | 0 | 0 |
| 0 | 0 | 223 | 168 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 151 |
| 0 | 0 | 0 | 0 | 188 | 234 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 3 |
| 0 | 0 | 0 | 0 | 160 | 185 |
G:=sub<GL(6,GF(241))| [190,51,0,0,0,0,190,240,0,0,0,0,0,0,0,240,0,0,0,0,1,240,0,0,0,0,0,0,64,101,0,0,0,0,0,177],[1,51,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,43,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,73,223,0,0,0,0,55,168,0,0,0,0,0,0,7,188,0,0,0,0,151,234],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,56,160,0,0,0,0,3,185] >;
C60.16C23 in GAP, Magma, Sage, TeX
C_{60}._{16}C_2^3 % in TeX
G:=Group("C60.16C2^3"); // GroupNames label
G:=SmallGroup(480,568);
// by ID
G=gap.SmallGroup(480,568);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,422,135,346,185,80,1356,18822]);
// 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,c*b*c^-1=d*b*d=a^30*b,d*c*d=a^45*c>;
// generators/relations