direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C23.D5, C10.62(C6×D4), (C23×C6).1D5, C24.2(C3×D5), (C22×C30)⋊14C4, (C2×C30).169D4, C30.419(C2×D4), (C23×C30).4C2, (C23×C10).4C6, C23⋊3(C3×Dic5), (C22×C6)⋊3Dic5, C22⋊3(C6×Dic5), C23.37(C6×D5), C30⋊11(C22⋊C4), (C22×C10)⋊10C12, C10.41(C22×C12), C30.226(C22×C4), (C2×C30).377C23, (C22×Dic5)⋊10C6, (C6×Dic5)⋊34C22, (C22×C6).134D10, C6.38(C22×Dic5), (C22×C30).162C22, C5⋊4(C6×C22⋊C4), C2.4(C6×C5⋊D4), (C2×C30)⋊42(C2×C4), C2.9(C2×C6×Dic5), C10⋊3(C3×C22⋊C4), C15⋊21(C2×C22⋊C4), (C2×C6×Dic5)⋊18C2, (C2×C6)⋊7(C2×Dic5), (C2×C10)⋊15(C2×C12), C22.27(D5×C2×C6), (C2×Dic5)⋊7(C2×C6), (C2×C10).44(C3×D4), C6.147(C2×C5⋊D4), (C2×C6).97(C5⋊D4), C22.25(C3×C5⋊D4), (C2×C10).60(C22×C6), (C22×C10).49(C2×C6), (C2×C6).373(C22×D5), SmallGroup(480,745)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C23.D5
G = < a,b,c,d,e,f | a6=b2=c2=d2=e5=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 592 in 264 conjugacy classes, 130 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C23, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C2×C10, C2×C10, C2×C10, C2×C12, C22×C6, C22×C6, C22×C6, C30, C30, C30, C2×C22⋊C4, C2×Dic5, C2×Dic5, C22×C10, C22×C10, C22×C10, C3×C22⋊C4, C22×C12, C23×C6, C3×Dic5, C2×C30, C2×C30, C2×C30, C23.D5, C22×Dic5, C23×C10, C6×C22⋊C4, C6×Dic5, C6×Dic5, C22×C30, C22×C30, C22×C30, C2×C23.D5, C3×C23.D5, C2×C6×Dic5, C23×C30, C6×C23.D5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, D5, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C2×C12, C3×D4, C22×C6, C3×D5, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, C3×C22⋊C4, C22×C12, C6×D4, C3×Dic5, C6×D5, C23.D5, C22×Dic5, C2×C5⋊D4, C6×C22⋊C4, C6×Dic5, C3×C5⋊D4, D5×C2×C6, C2×C23.D5, C3×C23.D5, C2×C6×Dic5, C6×C5⋊D4, C6×C23.D5
►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)
(19 205)(20 206)(21 207)(22 208)(23 209)(24 210)(37 225)(38 226)(39 227)(40 228)(41 223)(42 224)(73 135)(74 136)(75 137)(76 138)(77 133)(78 134)(91 196)(92 197)(93 198)(94 193)(95 194)(96 195)(115 188)(116 189)(117 190)(118 191)(119 192)(120 187)(121 229)(122 230)(123 231)(124 232)(125 233)(126 234)(127 217)(128 218)(129 219)(130 220)(131 221)(132 222)(145 235)(146 236)(147 237)(148 238)(149 239)(150 240)(163 213)(164 214)(165 215)(166 216)(167 211)(168 212)(169 201)(170 202)(171 203)(172 204)(173 199)(174 200)
(1 111)(2 112)(3 113)(4 114)(5 109)(6 110)(7 142)(8 143)(9 144)(10 139)(11 140)(12 141)(13 64)(14 65)(15 66)(16 61)(17 62)(18 63)(19 165)(20 166)(21 167)(22 168)(23 163)(24 164)(25 161)(26 162)(27 157)(28 158)(29 159)(30 160)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 125)(38 126)(39 121)(40 122)(41 123)(42 124)(43 177)(44 178)(45 179)(46 180)(47 175)(48 176)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(67 106)(68 107)(69 108)(70 103)(71 104)(72 105)(73 219)(74 220)(75 221)(76 222)(77 217)(78 218)(79 185)(80 186)(81 181)(82 182)(83 183)(84 184)(91 239)(92 240)(93 235)(94 236)(95 237)(96 238)(97 152)(98 153)(99 154)(100 155)(101 156)(102 151)(115 203)(116 204)(117 199)(118 200)(119 201)(120 202)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)(145 198)(146 193)(147 194)(148 195)(149 196)(150 197)(169 192)(170 187)(171 188)(172 189)(173 190)(174 191)(205 215)(206 216)(207 211)(208 212)(209 213)(210 214)(223 231)(224 232)(225 233)(226 234)(227 229)(228 230)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 99)(8 100)(9 101)(10 102)(11 97)(12 98)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 205)(20 206)(21 207)(22 208)(23 209)(24 210)(31 180)(32 175)(33 176)(34 177)(35 178)(36 179)(37 225)(38 226)(39 227)(40 228)(41 223)(42 224)(43 52)(44 53)(45 54)(46 49)(47 50)(48 51)(61 157)(62 158)(63 159)(64 160)(65 161)(66 162)(67 183)(68 184)(69 185)(70 186)(71 181)(72 182)(73 135)(74 136)(75 137)(76 138)(77 133)(78 134)(79 108)(80 103)(81 104)(82 105)(83 106)(84 107)(85 112)(86 113)(87 114)(88 109)(89 110)(90 111)(91 196)(92 197)(93 198)(94 193)(95 194)(96 195)(115 188)(116 189)(117 190)(118 191)(119 192)(120 187)(121 229)(122 230)(123 231)(124 232)(125 233)(126 234)(127 217)(128 218)(129 219)(130 220)(131 221)(132 222)(139 151)(140 152)(141 153)(142 154)(143 155)(144 156)(145 235)(146 236)(147 237)(148 238)(149 239)(150 240)(163 213)(164 214)(165 215)(166 216)(167 211)(168 212)(169 201)(170 202)(171 203)(172 204)(173 199)(174 200)
(1 17 83 35 154)(2 18 84 36 155)(3 13 79 31 156)(4 14 80 32 151)(5 15 81 33 152)(6 16 82 34 153)(7 90 158 67 44)(8 85 159 68 45)(9 86 160 69 46)(10 87 161 70 47)(11 88 162 71 48)(12 89 157 72 43)(19 174 146 121 136)(20 169 147 122 137)(21 170 148 123 138)(22 171 149 124 133)(23 172 150 125 134)(24 173 145 126 135)(25 103 175 139 57)(26 104 176 140 58)(27 105 177 141 59)(28 106 178 142 60)(29 107 179 143 55)(30 108 180 144 56)(37 128 163 189 197)(38 129 164 190 198)(39 130 165 191 193)(40 131 166 192 194)(41 132 167 187 195)(42 127 168 188 196)(49 101 113 64 185)(50 102 114 65 186)(51 97 109 66 181)(52 98 110 61 182)(53 99 111 62 183)(54 100 112 63 184)(73 210 199 235 234)(74 205 200 236 229)(75 206 201 237 230)(76 207 202 238 231)(77 208 203 239 232)(78 209 204 240 233)(91 224 217 212 115)(92 225 218 213 116)(93 226 219 214 117)(94 227 220 215 118)(95 228 221 216 119)(96 223 222 211 120)
(1 208 111 212)(2 209 112 213)(3 210 113 214)(4 205 114 215)(5 206 109 216)(6 207 110 211)(7 188 142 171)(8 189 143 172)(9 190 144 173)(10 191 139 174)(11 192 140 169)(12 187 141 170)(13 73 64 219)(14 74 65 220)(15 75 66 221)(16 76 61 222)(17 77 62 217)(18 78 63 218)(19 87 165 57)(20 88 166 58)(21 89 167 59)(22 90 168 60)(23 85 163 55)(24 86 164 56)(25 136 161 130)(26 137 162 131)(27 138 157 132)(28 133 158 127)(29 134 159 128)(30 135 160 129)(31 235 49 93)(32 236 50 94)(33 237 51 95)(34 238 52 96)(35 239 53 91)(36 240 54 92)(37 107 125 68)(38 108 126 69)(39 103 121 70)(40 104 122 71)(41 105 123 72)(42 106 124 67)(43 195 177 148)(44 196 178 149)(45 197 179 150)(46 198 180 145)(47 193 175 146)(48 194 176 147)(79 234 185 226)(80 229 186 227)(81 230 181 228)(82 231 182 223)(83 232 183 224)(84 233 184 225)(97 119 152 201)(98 120 153 202)(99 115 154 203)(100 116 155 204)(101 117 156 199)(102 118 151 200)
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), (19,205)(20,206)(21,207)(22,208)(23,209)(24,210)(37,225)(38,226)(39,227)(40,228)(41,223)(42,224)(73,135)(74,136)(75,137)(76,138)(77,133)(78,134)(91,196)(92,197)(93,198)(94,193)(95,194)(96,195)(115,188)(116,189)(117,190)(118,191)(119,192)(120,187)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(145,235)(146,236)(147,237)(148,238)(149,239)(150,240)(163,213)(164,214)(165,215)(166,216)(167,211)(168,212)(169,201)(170,202)(171,203)(172,204)(173,199)(174,200), (1,111)(2,112)(3,113)(4,114)(5,109)(6,110)(7,142)(8,143)(9,144)(10,139)(11,140)(12,141)(13,64)(14,65)(15,66)(16,61)(17,62)(18,63)(19,165)(20,166)(21,167)(22,168)(23,163)(24,164)(25,161)(26,162)(27,157)(28,158)(29,159)(30,160)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,125)(38,126)(39,121)(40,122)(41,123)(42,124)(43,177)(44,178)(45,179)(46,180)(47,175)(48,176)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(67,106)(68,107)(69,108)(70,103)(71,104)(72,105)(73,219)(74,220)(75,221)(76,222)(77,217)(78,218)(79,185)(80,186)(81,181)(82,182)(83,183)(84,184)(91,239)(92,240)(93,235)(94,236)(95,237)(96,238)(97,152)(98,153)(99,154)(100,155)(101,156)(102,151)(115,203)(116,204)(117,199)(118,200)(119,201)(120,202)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)(145,198)(146,193)(147,194)(148,195)(149,196)(150,197)(169,192)(170,187)(171,188)(172,189)(173,190)(174,191)(205,215)(206,216)(207,211)(208,212)(209,213)(210,214)(223,231)(224,232)(225,233)(226,234)(227,229)(228,230), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,99)(8,100)(9,101)(10,102)(11,97)(12,98)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(19,205)(20,206)(21,207)(22,208)(23,209)(24,210)(31,180)(32,175)(33,176)(34,177)(35,178)(36,179)(37,225)(38,226)(39,227)(40,228)(41,223)(42,224)(43,52)(44,53)(45,54)(46,49)(47,50)(48,51)(61,157)(62,158)(63,159)(64,160)(65,161)(66,162)(67,183)(68,184)(69,185)(70,186)(71,181)(72,182)(73,135)(74,136)(75,137)(76,138)(77,133)(78,134)(79,108)(80,103)(81,104)(82,105)(83,106)(84,107)(85,112)(86,113)(87,114)(88,109)(89,110)(90,111)(91,196)(92,197)(93,198)(94,193)(95,194)(96,195)(115,188)(116,189)(117,190)(118,191)(119,192)(120,187)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(139,151)(140,152)(141,153)(142,154)(143,155)(144,156)(145,235)(146,236)(147,237)(148,238)(149,239)(150,240)(163,213)(164,214)(165,215)(166,216)(167,211)(168,212)(169,201)(170,202)(171,203)(172,204)(173,199)(174,200), (1,17,83,35,154)(2,18,84,36,155)(3,13,79,31,156)(4,14,80,32,151)(5,15,81,33,152)(6,16,82,34,153)(7,90,158,67,44)(8,85,159,68,45)(9,86,160,69,46)(10,87,161,70,47)(11,88,162,71,48)(12,89,157,72,43)(19,174,146,121,136)(20,169,147,122,137)(21,170,148,123,138)(22,171,149,124,133)(23,172,150,125,134)(24,173,145,126,135)(25,103,175,139,57)(26,104,176,140,58)(27,105,177,141,59)(28,106,178,142,60)(29,107,179,143,55)(30,108,180,144,56)(37,128,163,189,197)(38,129,164,190,198)(39,130,165,191,193)(40,131,166,192,194)(41,132,167,187,195)(42,127,168,188,196)(49,101,113,64,185)(50,102,114,65,186)(51,97,109,66,181)(52,98,110,61,182)(53,99,111,62,183)(54,100,112,63,184)(73,210,199,235,234)(74,205,200,236,229)(75,206,201,237,230)(76,207,202,238,231)(77,208,203,239,232)(78,209,204,240,233)(91,224,217,212,115)(92,225,218,213,116)(93,226,219,214,117)(94,227,220,215,118)(95,228,221,216,119)(96,223,222,211,120), (1,208,111,212)(2,209,112,213)(3,210,113,214)(4,205,114,215)(5,206,109,216)(6,207,110,211)(7,188,142,171)(8,189,143,172)(9,190,144,173)(10,191,139,174)(11,192,140,169)(12,187,141,170)(13,73,64,219)(14,74,65,220)(15,75,66,221)(16,76,61,222)(17,77,62,217)(18,78,63,218)(19,87,165,57)(20,88,166,58)(21,89,167,59)(22,90,168,60)(23,85,163,55)(24,86,164,56)(25,136,161,130)(26,137,162,131)(27,138,157,132)(28,133,158,127)(29,134,159,128)(30,135,160,129)(31,235,49,93)(32,236,50,94)(33,237,51,95)(34,238,52,96)(35,239,53,91)(36,240,54,92)(37,107,125,68)(38,108,126,69)(39,103,121,70)(40,104,122,71)(41,105,123,72)(42,106,124,67)(43,195,177,148)(44,196,178,149)(45,197,179,150)(46,198,180,145)(47,193,175,146)(48,194,176,147)(79,234,185,226)(80,229,186,227)(81,230,181,228)(82,231,182,223)(83,232,183,224)(84,233,184,225)(97,119,152,201)(98,120,153,202)(99,115,154,203)(100,116,155,204)(101,117,156,199)(102,118,151,200)>;
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), (19,205)(20,206)(21,207)(22,208)(23,209)(24,210)(37,225)(38,226)(39,227)(40,228)(41,223)(42,224)(73,135)(74,136)(75,137)(76,138)(77,133)(78,134)(91,196)(92,197)(93,198)(94,193)(95,194)(96,195)(115,188)(116,189)(117,190)(118,191)(119,192)(120,187)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(145,235)(146,236)(147,237)(148,238)(149,239)(150,240)(163,213)(164,214)(165,215)(166,216)(167,211)(168,212)(169,201)(170,202)(171,203)(172,204)(173,199)(174,200), (1,111)(2,112)(3,113)(4,114)(5,109)(6,110)(7,142)(8,143)(9,144)(10,139)(11,140)(12,141)(13,64)(14,65)(15,66)(16,61)(17,62)(18,63)(19,165)(20,166)(21,167)(22,168)(23,163)(24,164)(25,161)(26,162)(27,157)(28,158)(29,159)(30,160)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,125)(38,126)(39,121)(40,122)(41,123)(42,124)(43,177)(44,178)(45,179)(46,180)(47,175)(48,176)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(67,106)(68,107)(69,108)(70,103)(71,104)(72,105)(73,219)(74,220)(75,221)(76,222)(77,217)(78,218)(79,185)(80,186)(81,181)(82,182)(83,183)(84,184)(91,239)(92,240)(93,235)(94,236)(95,237)(96,238)(97,152)(98,153)(99,154)(100,155)(101,156)(102,151)(115,203)(116,204)(117,199)(118,200)(119,201)(120,202)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)(145,198)(146,193)(147,194)(148,195)(149,196)(150,197)(169,192)(170,187)(171,188)(172,189)(173,190)(174,191)(205,215)(206,216)(207,211)(208,212)(209,213)(210,214)(223,231)(224,232)(225,233)(226,234)(227,229)(228,230), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,99)(8,100)(9,101)(10,102)(11,97)(12,98)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(19,205)(20,206)(21,207)(22,208)(23,209)(24,210)(31,180)(32,175)(33,176)(34,177)(35,178)(36,179)(37,225)(38,226)(39,227)(40,228)(41,223)(42,224)(43,52)(44,53)(45,54)(46,49)(47,50)(48,51)(61,157)(62,158)(63,159)(64,160)(65,161)(66,162)(67,183)(68,184)(69,185)(70,186)(71,181)(72,182)(73,135)(74,136)(75,137)(76,138)(77,133)(78,134)(79,108)(80,103)(81,104)(82,105)(83,106)(84,107)(85,112)(86,113)(87,114)(88,109)(89,110)(90,111)(91,196)(92,197)(93,198)(94,193)(95,194)(96,195)(115,188)(116,189)(117,190)(118,191)(119,192)(120,187)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(139,151)(140,152)(141,153)(142,154)(143,155)(144,156)(145,235)(146,236)(147,237)(148,238)(149,239)(150,240)(163,213)(164,214)(165,215)(166,216)(167,211)(168,212)(169,201)(170,202)(171,203)(172,204)(173,199)(174,200), (1,17,83,35,154)(2,18,84,36,155)(3,13,79,31,156)(4,14,80,32,151)(5,15,81,33,152)(6,16,82,34,153)(7,90,158,67,44)(8,85,159,68,45)(9,86,160,69,46)(10,87,161,70,47)(11,88,162,71,48)(12,89,157,72,43)(19,174,146,121,136)(20,169,147,122,137)(21,170,148,123,138)(22,171,149,124,133)(23,172,150,125,134)(24,173,145,126,135)(25,103,175,139,57)(26,104,176,140,58)(27,105,177,141,59)(28,106,178,142,60)(29,107,179,143,55)(30,108,180,144,56)(37,128,163,189,197)(38,129,164,190,198)(39,130,165,191,193)(40,131,166,192,194)(41,132,167,187,195)(42,127,168,188,196)(49,101,113,64,185)(50,102,114,65,186)(51,97,109,66,181)(52,98,110,61,182)(53,99,111,62,183)(54,100,112,63,184)(73,210,199,235,234)(74,205,200,236,229)(75,206,201,237,230)(76,207,202,238,231)(77,208,203,239,232)(78,209,204,240,233)(91,224,217,212,115)(92,225,218,213,116)(93,226,219,214,117)(94,227,220,215,118)(95,228,221,216,119)(96,223,222,211,120), (1,208,111,212)(2,209,112,213)(3,210,113,214)(4,205,114,215)(5,206,109,216)(6,207,110,211)(7,188,142,171)(8,189,143,172)(9,190,144,173)(10,191,139,174)(11,192,140,169)(12,187,141,170)(13,73,64,219)(14,74,65,220)(15,75,66,221)(16,76,61,222)(17,77,62,217)(18,78,63,218)(19,87,165,57)(20,88,166,58)(21,89,167,59)(22,90,168,60)(23,85,163,55)(24,86,164,56)(25,136,161,130)(26,137,162,131)(27,138,157,132)(28,133,158,127)(29,134,159,128)(30,135,160,129)(31,235,49,93)(32,236,50,94)(33,237,51,95)(34,238,52,96)(35,239,53,91)(36,240,54,92)(37,107,125,68)(38,108,126,69)(39,103,121,70)(40,104,122,71)(41,105,123,72)(42,106,124,67)(43,195,177,148)(44,196,178,149)(45,197,179,150)(46,198,180,145)(47,193,175,146)(48,194,176,147)(79,234,185,226)(80,229,186,227)(81,230,181,228)(82,231,182,223)(83,232,183,224)(84,233,184,225)(97,119,152,201)(98,120,153,202)(99,115,154,203)(100,116,155,204)(101,117,156,199)(102,118,151,200) );
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)], [(19,205),(20,206),(21,207),(22,208),(23,209),(24,210),(37,225),(38,226),(39,227),(40,228),(41,223),(42,224),(73,135),(74,136),(75,137),(76,138),(77,133),(78,134),(91,196),(92,197),(93,198),(94,193),(95,194),(96,195),(115,188),(116,189),(117,190),(118,191),(119,192),(120,187),(121,229),(122,230),(123,231),(124,232),(125,233),(126,234),(127,217),(128,218),(129,219),(130,220),(131,221),(132,222),(145,235),(146,236),(147,237),(148,238),(149,239),(150,240),(163,213),(164,214),(165,215),(166,216),(167,211),(168,212),(169,201),(170,202),(171,203),(172,204),(173,199),(174,200)], [(1,111),(2,112),(3,113),(4,114),(5,109),(6,110),(7,142),(8,143),(9,144),(10,139),(11,140),(12,141),(13,64),(14,65),(15,66),(16,61),(17,62),(18,63),(19,165),(20,166),(21,167),(22,168),(23,163),(24,164),(25,161),(26,162),(27,157),(28,158),(29,159),(30,160),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,125),(38,126),(39,121),(40,122),(41,123),(42,124),(43,177),(44,178),(45,179),(46,180),(47,175),(48,176),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(67,106),(68,107),(69,108),(70,103),(71,104),(72,105),(73,219),(74,220),(75,221),(76,222),(77,217),(78,218),(79,185),(80,186),(81,181),(82,182),(83,183),(84,184),(91,239),(92,240),(93,235),(94,236),(95,237),(96,238),(97,152),(98,153),(99,154),(100,155),(101,156),(102,151),(115,203),(116,204),(117,199),(118,200),(119,201),(120,202),(127,133),(128,134),(129,135),(130,136),(131,137),(132,138),(145,198),(146,193),(147,194),(148,195),(149,196),(150,197),(169,192),(170,187),(171,188),(172,189),(173,190),(174,191),(205,215),(206,216),(207,211),(208,212),(209,213),(210,214),(223,231),(224,232),(225,233),(226,234),(227,229),(228,230)], [(1,60),(2,55),(3,56),(4,57),(5,58),(6,59),(7,99),(8,100),(9,101),(10,102),(11,97),(12,98),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(19,205),(20,206),(21,207),(22,208),(23,209),(24,210),(31,180),(32,175),(33,176),(34,177),(35,178),(36,179),(37,225),(38,226),(39,227),(40,228),(41,223),(42,224),(43,52),(44,53),(45,54),(46,49),(47,50),(48,51),(61,157),(62,158),(63,159),(64,160),(65,161),(66,162),(67,183),(68,184),(69,185),(70,186),(71,181),(72,182),(73,135),(74,136),(75,137),(76,138),(77,133),(78,134),(79,108),(80,103),(81,104),(82,105),(83,106),(84,107),(85,112),(86,113),(87,114),(88,109),(89,110),(90,111),(91,196),(92,197),(93,198),(94,193),(95,194),(96,195),(115,188),(116,189),(117,190),(118,191),(119,192),(120,187),(121,229),(122,230),(123,231),(124,232),(125,233),(126,234),(127,217),(128,218),(129,219),(130,220),(131,221),(132,222),(139,151),(140,152),(141,153),(142,154),(143,155),(144,156),(145,235),(146,236),(147,237),(148,238),(149,239),(150,240),(163,213),(164,214),(165,215),(166,216),(167,211),(168,212),(169,201),(170,202),(171,203),(172,204),(173,199),(174,200)], [(1,17,83,35,154),(2,18,84,36,155),(3,13,79,31,156),(4,14,80,32,151),(5,15,81,33,152),(6,16,82,34,153),(7,90,158,67,44),(8,85,159,68,45),(9,86,160,69,46),(10,87,161,70,47),(11,88,162,71,48),(12,89,157,72,43),(19,174,146,121,136),(20,169,147,122,137),(21,170,148,123,138),(22,171,149,124,133),(23,172,150,125,134),(24,173,145,126,135),(25,103,175,139,57),(26,104,176,140,58),(27,105,177,141,59),(28,106,178,142,60),(29,107,179,143,55),(30,108,180,144,56),(37,128,163,189,197),(38,129,164,190,198),(39,130,165,191,193),(40,131,166,192,194),(41,132,167,187,195),(42,127,168,188,196),(49,101,113,64,185),(50,102,114,65,186),(51,97,109,66,181),(52,98,110,61,182),(53,99,111,62,183),(54,100,112,63,184),(73,210,199,235,234),(74,205,200,236,229),(75,206,201,237,230),(76,207,202,238,231),(77,208,203,239,232),(78,209,204,240,233),(91,224,217,212,115),(92,225,218,213,116),(93,226,219,214,117),(94,227,220,215,118),(95,228,221,216,119),(96,223,222,211,120)], [(1,208,111,212),(2,209,112,213),(3,210,113,214),(4,205,114,215),(5,206,109,216),(6,207,110,211),(7,188,142,171),(8,189,143,172),(9,190,144,173),(10,191,139,174),(11,192,140,169),(12,187,141,170),(13,73,64,219),(14,74,65,220),(15,75,66,221),(16,76,61,222),(17,77,62,217),(18,78,63,218),(19,87,165,57),(20,88,166,58),(21,89,167,59),(22,90,168,60),(23,85,163,55),(24,86,164,56),(25,136,161,130),(26,137,162,131),(27,138,157,132),(28,133,158,127),(29,134,159,128),(30,135,160,129),(31,235,49,93),(32,236,50,94),(33,237,51,95),(34,238,52,96),(35,239,53,91),(36,240,54,92),(37,107,125,68),(38,108,126,69),(39,103,121,70),(40,104,122,71),(41,105,123,72),(42,106,124,67),(43,195,177,148),(44,196,178,149),(45,197,179,150),(46,198,180,145),(47,193,175,146),(48,194,176,147),(79,234,185,226),(80,229,186,227),(81,230,181,228),(82,231,182,223),(83,232,183,224),(84,233,184,225),(97,119,152,201),(98,120,153,202),(99,115,154,203),(100,116,155,204),(101,117,156,199),(102,118,151,200)]])
156 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 5A | 5B | 6A | ··· | 6N | 6O | ··· | 6V | 10A | ··· | 10AD | 12A | ··· | 12P | 15A | 15B | 15C | 15D | 30A | ··· | 30BH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D5 | Dic5 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C3×Dic5 | C6×D5 | C3×C5⋊D4 |
| kernel | C6×C23.D5 | C3×C23.D5 | C2×C6×Dic5 | C23×C30 | C2×C23.D5 | C22×C30 | C23.D5 | C22×Dic5 | C23×C10 | C22×C10 | C2×C30 | C23×C6 | C22×C6 | C22×C6 | C2×C10 | C24 | C2×C6 | C23 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 2 | 16 | 4 | 2 | 8 | 6 | 8 | 4 | 16 | 16 | 12 | 32 |
Matrix representation of C6×C23.D5 ►in GL4(𝔽61) generated by
| 47 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 27 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 36 | 58 |
| 11 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 48 | 10 |
| 0 | 0 | 32 | 13 |
G:=sub<GL(4,GF(61))| [47,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,27,0,0,0,60],[60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,20,36,0,0,0,58],[11,0,0,0,0,1,0,0,0,0,48,32,0,0,10,13] >;
C6×C23.D5 in GAP, Magma, Sage, TeX
C_6\times C_2^3.D_5
% in TeX
G:=Group("C6xC2^3.D5"); // GroupNames label
G:=SmallGroup(480,745);
// by ID
G=gap.SmallGroup(480,745);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^6=b^2=c^2=d^2=e^5=1,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations