metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.19C23, Dic6.9D10, D60.5C22, Dic10.24D6, D4⋊D15⋊7C2, D4.D5⋊7S3, D4.5(S3×D5), D4⋊2S3⋊3D5, C15⋊16(C4○D8), (C5×D4).22D6, C15⋊Q16⋊4C2, (C3×D4).7D10, C5⋊2C8.14D6, D60⋊C2⋊1C2, (S3×C10).11D4, (C4×S3).22D10, C10.145(S3×D4), C30.181(C2×D4), Dic6⋊D5⋊3C2, D6.2(C5⋊D4), C5⋊6(Q8.7D6), C3⋊3(D4.8D10), (S3×C20).7C22, C20.19(C22×S3), C15⋊3C8.6C22, (C5×Dic3).37D4, C12.19(C22×D5), (D4×C15).13C22, (C5×Dic6).6C22, Dic3.21(C5⋊D4), (C3×Dic10).6C22, C4.19(C2×S3×D5), (S3×C5⋊2C8)⋊4C2, (C3×D4.D5)⋊5C2, C2.26(S3×C5⋊D4), C6.48(C2×C5⋊D4), (C5×D4⋊2S3)⋊3C2, (C3×C5⋊2C8).4C22, SmallGroup(480,571)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.19C23
G = < a,b,c,d | a60=d2=1, b2=c2=a30, bab-1=a19, cac-1=a41, dad=a31, bc=cb, dbd=a45b, cd=dc >
Subgroups: 636 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8 [×2], C2×C4 [×3], D4, D4 [×3], Q8 [×2], D5, C10, C10 [×2], Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10, C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12 [×2], C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, D15, C30, C30, C4○D8, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C5×Dic3, C5×Dic3, C3×Dic5, C60, S3×C10, D30, C2×C30, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, Q8.7D6, C3×C5⋊2C8, C15⋊3C8, D30.C2, C5⋊D12, C3×Dic10, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, D60, D4×C15, D4.8D10, S3×C5⋊2C8, Dic6⋊D5, C15⋊Q16, C3×D4.D5, D4⋊D15, D60⋊C2, C5×D4⋊2S3, C60.19C23
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q8.7D6, C2×S3×D5, D4.8D10, S3×C5⋊D4, C60.19C23
►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)
(1 137 31 167)(2 156 32 126)(3 175 33 145)(4 134 34 164)(5 153 35 123)(6 172 36 142)(7 131 37 161)(8 150 38 180)(9 169 39 139)(10 128 40 158)(11 147 41 177)(12 166 42 136)(13 125 43 155)(14 144 44 174)(15 163 45 133)(16 122 46 152)(17 141 47 171)(18 160 48 130)(19 179 49 149)(20 138 50 168)(21 157 51 127)(22 176 52 146)(23 135 53 165)(24 154 54 124)(25 173 55 143)(26 132 56 162)(27 151 57 121)(28 170 58 140)(29 129 59 159)(30 148 60 178)(61 202 91 232)(62 221 92 191)(63 240 93 210)(64 199 94 229)(65 218 95 188)(66 237 96 207)(67 196 97 226)(68 215 98 185)(69 234 99 204)(70 193 100 223)(71 212 101 182)(72 231 102 201)(73 190 103 220)(74 209 104 239)(75 228 105 198)(76 187 106 217)(77 206 107 236)(78 225 108 195)(79 184 109 214)(80 203 110 233)(81 222 111 192)(82 181 112 211)(83 200 113 230)(84 219 114 189)(85 238 115 208)(86 197 116 227)(87 216 117 186)(88 235 118 205)(89 194 119 224)(90 213 120 183)
(1 16 31 46)(2 57 32 27)(3 38 33 8)(4 19 34 49)(5 60 35 30)(6 41 36 11)(7 22 37 52)(9 44 39 14)(10 25 40 55)(12 47 42 17)(13 28 43 58)(15 50 45 20)(18 53 48 23)(21 56 51 26)(24 59 54 29)(61 66 91 96)(62 107 92 77)(63 88 93 118)(64 69 94 99)(65 110 95 80)(67 72 97 102)(68 113 98 83)(70 75 100 105)(71 116 101 86)(73 78 103 108)(74 119 104 89)(76 81 106 111)(79 84 109 114)(82 87 112 117)(85 90 115 120)(121 126 151 156)(122 167 152 137)(123 148 153 178)(124 129 154 159)(125 170 155 140)(127 132 157 162)(128 173 158 143)(130 135 160 165)(131 176 161 146)(133 138 163 168)(134 179 164 149)(136 141 166 171)(139 144 169 174)(142 147 172 177)(145 150 175 180)(181 216 211 186)(182 197 212 227)(183 238 213 208)(184 219 214 189)(185 200 215 230)(187 222 217 192)(188 203 218 233)(190 225 220 195)(191 206 221 236)(193 228 223 198)(194 209 224 239)(196 231 226 201)(199 234 229 204)(202 237 232 207)(205 240 235 210)
(1 98)(2 69)(3 100)(4 71)(5 102)(6 73)(7 104)(8 75)(9 106)(10 77)(11 108)(12 79)(13 110)(14 81)(15 112)(16 83)(17 114)(18 85)(19 116)(20 87)(21 118)(22 89)(23 120)(24 91)(25 62)(26 93)(27 64)(28 95)(29 66)(30 97)(31 68)(32 99)(33 70)(34 101)(35 72)(36 103)(37 74)(38 105)(39 76)(40 107)(41 78)(42 109)(43 80)(44 111)(45 82)(46 113)(47 84)(48 115)(49 86)(50 117)(51 88)(52 119)(53 90)(54 61)(55 92)(56 63)(57 94)(58 65)(59 96)(60 67)(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), (1,137,31,167)(2,156,32,126)(3,175,33,145)(4,134,34,164)(5,153,35,123)(6,172,36,142)(7,131,37,161)(8,150,38,180)(9,169,39,139)(10,128,40,158)(11,147,41,177)(12,166,42,136)(13,125,43,155)(14,144,44,174)(15,163,45,133)(16,122,46,152)(17,141,47,171)(18,160,48,130)(19,179,49,149)(20,138,50,168)(21,157,51,127)(22,176,52,146)(23,135,53,165)(24,154,54,124)(25,173,55,143)(26,132,56,162)(27,151,57,121)(28,170,58,140)(29,129,59,159)(30,148,60,178)(61,202,91,232)(62,221,92,191)(63,240,93,210)(64,199,94,229)(65,218,95,188)(66,237,96,207)(67,196,97,226)(68,215,98,185)(69,234,99,204)(70,193,100,223)(71,212,101,182)(72,231,102,201)(73,190,103,220)(74,209,104,239)(75,228,105,198)(76,187,106,217)(77,206,107,236)(78,225,108,195)(79,184,109,214)(80,203,110,233)(81,222,111,192)(82,181,112,211)(83,200,113,230)(84,219,114,189)(85,238,115,208)(86,197,116,227)(87,216,117,186)(88,235,118,205)(89,194,119,224)(90,213,120,183), (1,16,31,46)(2,57,32,27)(3,38,33,8)(4,19,34,49)(5,60,35,30)(6,41,36,11)(7,22,37,52)(9,44,39,14)(10,25,40,55)(12,47,42,17)(13,28,43,58)(15,50,45,20)(18,53,48,23)(21,56,51,26)(24,59,54,29)(61,66,91,96)(62,107,92,77)(63,88,93,118)(64,69,94,99)(65,110,95,80)(67,72,97,102)(68,113,98,83)(70,75,100,105)(71,116,101,86)(73,78,103,108)(74,119,104,89)(76,81,106,111)(79,84,109,114)(82,87,112,117)(85,90,115,120)(121,126,151,156)(122,167,152,137)(123,148,153,178)(124,129,154,159)(125,170,155,140)(127,132,157,162)(128,173,158,143)(130,135,160,165)(131,176,161,146)(133,138,163,168)(134,179,164,149)(136,141,166,171)(139,144,169,174)(142,147,172,177)(145,150,175,180)(181,216,211,186)(182,197,212,227)(183,238,213,208)(184,219,214,189)(185,200,215,230)(187,222,217,192)(188,203,218,233)(190,225,220,195)(191,206,221,236)(193,228,223,198)(194,209,224,239)(196,231,226,201)(199,234,229,204)(202,237,232,207)(205,240,235,210), (1,98)(2,69)(3,100)(4,71)(5,102)(6,73)(7,104)(8,75)(9,106)(10,77)(11,108)(12,79)(13,110)(14,81)(15,112)(16,83)(17,114)(18,85)(19,116)(20,87)(21,118)(22,89)(23,120)(24,91)(25,62)(26,93)(27,64)(28,95)(29,66)(30,97)(31,68)(32,99)(33,70)(34,101)(35,72)(36,103)(37,74)(38,105)(39,76)(40,107)(41,78)(42,109)(43,80)(44,111)(45,82)(46,113)(47,84)(48,115)(49,86)(50,117)(51,88)(52,119)(53,90)(54,61)(55,92)(56,63)(57,94)(58,65)(59,96)(60,67)(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), (1,137,31,167)(2,156,32,126)(3,175,33,145)(4,134,34,164)(5,153,35,123)(6,172,36,142)(7,131,37,161)(8,150,38,180)(9,169,39,139)(10,128,40,158)(11,147,41,177)(12,166,42,136)(13,125,43,155)(14,144,44,174)(15,163,45,133)(16,122,46,152)(17,141,47,171)(18,160,48,130)(19,179,49,149)(20,138,50,168)(21,157,51,127)(22,176,52,146)(23,135,53,165)(24,154,54,124)(25,173,55,143)(26,132,56,162)(27,151,57,121)(28,170,58,140)(29,129,59,159)(30,148,60,178)(61,202,91,232)(62,221,92,191)(63,240,93,210)(64,199,94,229)(65,218,95,188)(66,237,96,207)(67,196,97,226)(68,215,98,185)(69,234,99,204)(70,193,100,223)(71,212,101,182)(72,231,102,201)(73,190,103,220)(74,209,104,239)(75,228,105,198)(76,187,106,217)(77,206,107,236)(78,225,108,195)(79,184,109,214)(80,203,110,233)(81,222,111,192)(82,181,112,211)(83,200,113,230)(84,219,114,189)(85,238,115,208)(86,197,116,227)(87,216,117,186)(88,235,118,205)(89,194,119,224)(90,213,120,183), (1,16,31,46)(2,57,32,27)(3,38,33,8)(4,19,34,49)(5,60,35,30)(6,41,36,11)(7,22,37,52)(9,44,39,14)(10,25,40,55)(12,47,42,17)(13,28,43,58)(15,50,45,20)(18,53,48,23)(21,56,51,26)(24,59,54,29)(61,66,91,96)(62,107,92,77)(63,88,93,118)(64,69,94,99)(65,110,95,80)(67,72,97,102)(68,113,98,83)(70,75,100,105)(71,116,101,86)(73,78,103,108)(74,119,104,89)(76,81,106,111)(79,84,109,114)(82,87,112,117)(85,90,115,120)(121,126,151,156)(122,167,152,137)(123,148,153,178)(124,129,154,159)(125,170,155,140)(127,132,157,162)(128,173,158,143)(130,135,160,165)(131,176,161,146)(133,138,163,168)(134,179,164,149)(136,141,166,171)(139,144,169,174)(142,147,172,177)(145,150,175,180)(181,216,211,186)(182,197,212,227)(183,238,213,208)(184,219,214,189)(185,200,215,230)(187,222,217,192)(188,203,218,233)(190,225,220,195)(191,206,221,236)(193,228,223,198)(194,209,224,239)(196,231,226,201)(199,234,229,204)(202,237,232,207)(205,240,235,210), (1,98)(2,69)(3,100)(4,71)(5,102)(6,73)(7,104)(8,75)(9,106)(10,77)(11,108)(12,79)(13,110)(14,81)(15,112)(16,83)(17,114)(18,85)(19,116)(20,87)(21,118)(22,89)(23,120)(24,91)(25,62)(26,93)(27,64)(28,95)(29,66)(30,97)(31,68)(32,99)(33,70)(34,101)(35,72)(36,103)(37,74)(38,105)(39,76)(40,107)(41,78)(42,109)(43,80)(44,111)(45,82)(46,113)(47,84)(48,115)(49,86)(50,117)(51,88)(52,119)(53,90)(54,61)(55,92)(56,63)(57,94)(58,65)(59,96)(60,67)(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)], [(1,137,31,167),(2,156,32,126),(3,175,33,145),(4,134,34,164),(5,153,35,123),(6,172,36,142),(7,131,37,161),(8,150,38,180),(9,169,39,139),(10,128,40,158),(11,147,41,177),(12,166,42,136),(13,125,43,155),(14,144,44,174),(15,163,45,133),(16,122,46,152),(17,141,47,171),(18,160,48,130),(19,179,49,149),(20,138,50,168),(21,157,51,127),(22,176,52,146),(23,135,53,165),(24,154,54,124),(25,173,55,143),(26,132,56,162),(27,151,57,121),(28,170,58,140),(29,129,59,159),(30,148,60,178),(61,202,91,232),(62,221,92,191),(63,240,93,210),(64,199,94,229),(65,218,95,188),(66,237,96,207),(67,196,97,226),(68,215,98,185),(69,234,99,204),(70,193,100,223),(71,212,101,182),(72,231,102,201),(73,190,103,220),(74,209,104,239),(75,228,105,198),(76,187,106,217),(77,206,107,236),(78,225,108,195),(79,184,109,214),(80,203,110,233),(81,222,111,192),(82,181,112,211),(83,200,113,230),(84,219,114,189),(85,238,115,208),(86,197,116,227),(87,216,117,186),(88,235,118,205),(89,194,119,224),(90,213,120,183)], [(1,16,31,46),(2,57,32,27),(3,38,33,8),(4,19,34,49),(5,60,35,30),(6,41,36,11),(7,22,37,52),(9,44,39,14),(10,25,40,55),(12,47,42,17),(13,28,43,58),(15,50,45,20),(18,53,48,23),(21,56,51,26),(24,59,54,29),(61,66,91,96),(62,107,92,77),(63,88,93,118),(64,69,94,99),(65,110,95,80),(67,72,97,102),(68,113,98,83),(70,75,100,105),(71,116,101,86),(73,78,103,108),(74,119,104,89),(76,81,106,111),(79,84,109,114),(82,87,112,117),(85,90,115,120),(121,126,151,156),(122,167,152,137),(123,148,153,178),(124,129,154,159),(125,170,155,140),(127,132,157,162),(128,173,158,143),(130,135,160,165),(131,176,161,146),(133,138,163,168),(134,179,164,149),(136,141,166,171),(139,144,169,174),(142,147,172,177),(145,150,175,180),(181,216,211,186),(182,197,212,227),(183,238,213,208),(184,219,214,189),(185,200,215,230),(187,222,217,192),(188,203,218,233),(190,225,220,195),(191,206,221,236),(193,228,223,198),(194,209,224,239),(196,231,226,201),(199,234,229,204),(202,237,232,207),(205,240,235,210)], [(1,98),(2,69),(3,100),(4,71),(5,102),(6,73),(7,104),(8,75),(9,106),(10,77),(11,108),(12,79),(13,110),(14,81),(15,112),(16,83),(17,114),(18,85),(19,116),(20,87),(21,118),(22,89),(23,120),(24,91),(25,62),(26,93),(27,64),(28,95),(29,66),(30,97),(31,68),(32,99),(33,70),(34,101),(35,72),(36,103),(37,74),(38,105),(39,76),(40,107),(41,78),(42,109),(43,80),(44,111),(45,82),(46,113),(47,84),(48,115),(49,86),(50,117),(51,88),(52,119),(53,90),(54,61),(55,92),(56,63),(57,94),(58,65),(59,96),(60,67),(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)])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 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 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 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 | 60 | 2 | 2 | 3 | 3 | 12 | 20 | 2 | 2 | 2 | 8 | 10 | 10 | 30 | 30 | 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 |
51 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 | C4○D8 | C5⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | Q8.7D6 | C2×S3×D5 | D4.8D10 | S3×C5⋊D4 | C60.19C23 |
| kernel | C60.19C23 | S3×C5⋊2C8 | Dic6⋊D5 | C15⋊Q16 | C3×D4.D5 | D4⋊D15 | D60⋊C2 | C5×D4⋊2S3 | D4.D5 | C5×Dic3 | S3×C10 | D4⋊2S3 | C5⋊2C8 | Dic10 | C5×D4 | Dic6 | C4×S3 | C3×D4 | C15 | Dic3 | D6 | 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 | 4 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of C60.19C23 ►in GL6(𝔽241)
| 52 | 52 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 54 | 0 | 0 |
| 0 | 0 | 174 | 239 | 0 | 0 |
| 0 | 0 | 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 0 | 18 | 64 |
| 40 | 146 | 0 | 0 | 0 | 0 |
| 235 | 201 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 154 | 24 |
| 0 | 0 | 0 | 0 | 26 | 87 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 174 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 200 | 3 |
| 0 | 0 | 0 | 0 | 163 | 41 |
G:=sub<GL(6,GF(241))| [52,189,0,0,0,0,52,240,0,0,0,0,0,0,1,174,0,0,0,0,54,239,0,0,0,0,0,0,177,18,0,0,0,0,0,64],[40,235,0,0,0,0,146,201,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,154,26,0,0,0,0,24,87],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,174,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,200,163,0,0,0,0,3,41] >;
C60.19C23 in GAP, Magma, Sage, TeX
C_{60}._{19}C_2^3 % in TeX
G:=Group("C60.19C2^3"); // GroupNames label
G:=SmallGroup(480,571);
// by ID
G=gap.SmallGroup(480,571);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^60=d^2=1,b^2=c^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^41,d*a*d=a^31,b*c=c*b,d*b*d=a^45*b,c*d=d*c>;
// generators/relations