direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5×Dic6, C30.1C24, C60.135C23, Dic30⋊34C22, Dic15.1C23, C6⋊2(Q8×D5), (C6×D5)⋊8Q8, C30⋊1(C2×Q8), C15⋊Q8⋊6C22, C15⋊1(C22×Q8), (C4×D5).82D6, C10⋊1(C2×Dic6), C6.1(C23×D5), C5⋊1(C22×Dic6), (C10×Dic6)⋊7C2, (C2×C20).161D6, C10.1(S3×C23), (C2×Dic30)⋊29C2, (C2×C12).309D10, (C6×D5).37C23, (C2×C30).220C23, (C2×C60).153C22, C20.121(C22×S3), (C2×Dic5).197D6, (C5×Dic6)⋊20C22, (C22×D5).111D6, D10.52(C22×S3), (D5×C12).96C22, C12.158(C22×D5), Dic3.1(C22×D5), (D5×Dic3).9C22, (C5×Dic3).1C23, (C2×Dic3).128D10, (C3×Dic5).39C23, Dic5.41(C22×S3), (C6×Dic5).226C22, (C10×Dic3).126C22, (C2×Dic15).147C22, C3⋊2(C2×Q8×D5), (C2×C4×D5).6S3, C4.83(C2×S3×D5), (C2×C15⋊Q8)⋊19C2, (C3×D5)⋊1(C2×Q8), (D5×C2×C12).6C2, C2.5(C22×S3×D5), (C2×D5×Dic3).9C2, C22.92(C2×S3×D5), (C2×C4).166(S3×D5), (D5×C2×C6).114C22, (C2×C6).232(C22×D5), (C2×C10).232(C22×S3), SmallGroup(480,1073)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D5×Dic6
G = < a,b,c,d,e | a2=b5=c2=d12=1, e2=d6, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 1308 in 312 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C15, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C22×Q8, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C2×C30, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, Q8×C10, C22×Dic6, D5×Dic3, C15⋊Q8, D5×C12, C6×Dic5, C5×Dic6, C10×Dic3, Dic30, C2×Dic15, C2×C60, D5×C2×C6, C2×Q8×D5, D5×Dic6, C2×D5×Dic3, C2×C15⋊Q8, D5×C2×C12, C10×Dic6, C2×Dic30, C2×D5×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C24, D10, Dic6, C22×S3, C22×Q8, C22×D5, C2×Dic6, S3×C23, S3×D5, Q8×D5, C23×D5, C22×Dic6, C2×S3×D5, C2×Q8×D5, D5×Dic6, C22×S3×D5, C2×D5×Dic6
►On 240 points
Generators in S240
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 187)(14 188)(15 189)(16 190)(17 191)(18 192)(19 181)(20 182)(21 183)(22 184)(23 185)(24 186)(25 153)(26 154)(27 155)(28 156)(29 145)(30 146)(31 147)(32 148)(33 149)(34 150)(35 151)(36 152)(49 171)(50 172)(51 173)(52 174)(53 175)(54 176)(55 177)(56 178)(57 179)(58 180)(59 169)(60 170)(61 228)(62 217)(63 218)(64 219)(65 220)(66 221)(67 222)(68 223)(69 224)(70 225)(71 226)(72 227)(73 236)(74 237)(75 238)(76 239)(77 240)(78 229)(79 230)(80 231)(81 232)(82 233)(83 234)(84 235)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 193)(94 194)(95 195)(96 196)(97 111)(98 112)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 119)(106 120)(107 109)(108 110)(121 213)(122 214)(123 215)(124 216)(125 205)(126 206)(127 207)(128 208)(129 209)(130 210)(131 211)(132 212)(133 165)(134 166)(135 167)(136 168)(137 157)(138 158)(139 159)(140 160)(141 161)(142 162)(143 163)(144 164)
(1 52 22 216 111)(2 53 23 205 112)(3 54 24 206 113)(4 55 13 207 114)(5 56 14 208 115)(6 57 15 209 116)(7 58 16 210 117)(8 59 17 211 118)(9 60 18 212 119)(10 49 19 213 120)(11 50 20 214 109)(12 51 21 215 110)(25 61 78 87 141)(26 62 79 88 142)(27 63 80 89 143)(28 64 81 90 144)(29 65 82 91 133)(30 66 83 92 134)(31 67 84 93 135)(32 68 73 94 136)(33 69 74 95 137)(34 70 75 96 138)(35 71 76 85 139)(36 72 77 86 140)(37 170 192 132 105)(38 171 181 121 106)(39 172 182 122 107)(40 173 183 123 108)(41 174 184 124 97)(42 175 185 125 98)(43 176 186 126 99)(44 177 187 127 100)(45 178 188 128 101)(46 179 189 129 102)(47 180 190 130 103)(48 169 191 131 104)(145 220 233 203 165)(146 221 234 204 166)(147 222 235 193 167)(148 223 236 194 168)(149 224 237 195 157)(150 225 238 196 158)(151 226 239 197 159)(152 227 240 198 160)(153 228 229 199 161)(154 217 230 200 162)(155 218 231 201 163)(156 219 232 202 164)
(1 117)(2 118)(3 119)(4 120)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 133)(36 134)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 97)(48 98)(49 207)(50 208)(51 209)(52 210)(53 211)(54 212)(55 213)(56 214)(57 215)(58 216)(59 205)(60 206)(61 93)(62 94)(63 95)(64 96)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(121 177)(122 178)(123 179)(124 180)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(155 157)(156 158)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)(193 228)(194 217)(195 218)(196 219)(197 220)(198 221)(199 222)(200 223)(201 224)(202 225)(203 226)(204 227)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)
(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)
(1 154 7 148)(2 153 8 147)(3 152 9 146)(4 151 10 145)(5 150 11 156)(6 149 12 155)(13 239 19 233)(14 238 20 232)(15 237 21 231)(16 236 22 230)(17 235 23 229)(18 234 24 240)(25 48 31 42)(26 47 32 41)(27 46 33 40)(28 45 34 39)(29 44 35 38)(30 43 36 37)(49 220 55 226)(50 219 56 225)(51 218 57 224)(52 217 58 223)(53 228 59 222)(54 227 60 221)(61 169 67 175)(62 180 68 174)(63 179 69 173)(64 178 70 172)(65 177 71 171)(66 176 72 170)(73 184 79 190)(74 183 80 189)(75 182 81 188)(76 181 82 187)(77 192 83 186)(78 191 84 185)(85 121 91 127)(86 132 92 126)(87 131 93 125)(88 130 94 124)(89 129 95 123)(90 128 96 122)(97 142 103 136)(98 141 104 135)(99 140 105 134)(100 139 106 133)(101 138 107 144)(102 137 108 143)(109 164 115 158)(110 163 116 157)(111 162 117 168)(112 161 118 167)(113 160 119 166)(114 159 120 165)(193 205 199 211)(194 216 200 210)(195 215 201 209)(196 214 202 208)(197 213 203 207)(198 212 204 206)
G:=sub<Sym(240)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,187)(14,188)(15,189)(16,190)(17,191)(18,192)(19,181)(20,182)(21,183)(22,184)(23,185)(24,186)(25,153)(26,154)(27,155)(28,156)(29,145)(30,146)(31,147)(32,148)(33,149)(34,150)(35,151)(36,152)(49,171)(50,172)(51,173)(52,174)(53,175)(54,176)(55,177)(56,178)(57,179)(58,180)(59,169)(60,170)(61,228)(62,217)(63,218)(64,219)(65,220)(66,221)(67,222)(68,223)(69,224)(70,225)(71,226)(72,227)(73,236)(74,237)(75,238)(76,239)(77,240)(78,229)(79,230)(80,231)(81,232)(82,233)(83,234)(84,235)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,193)(94,194)(95,195)(96,196)(97,111)(98,112)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,109)(108,110)(121,213)(122,214)(123,215)(124,216)(125,205)(126,206)(127,207)(128,208)(129,209)(130,210)(131,211)(132,212)(133,165)(134,166)(135,167)(136,168)(137,157)(138,158)(139,159)(140,160)(141,161)(142,162)(143,163)(144,164), (1,52,22,216,111)(2,53,23,205,112)(3,54,24,206,113)(4,55,13,207,114)(5,56,14,208,115)(6,57,15,209,116)(7,58,16,210,117)(8,59,17,211,118)(9,60,18,212,119)(10,49,19,213,120)(11,50,20,214,109)(12,51,21,215,110)(25,61,78,87,141)(26,62,79,88,142)(27,63,80,89,143)(28,64,81,90,144)(29,65,82,91,133)(30,66,83,92,134)(31,67,84,93,135)(32,68,73,94,136)(33,69,74,95,137)(34,70,75,96,138)(35,71,76,85,139)(36,72,77,86,140)(37,170,192,132,105)(38,171,181,121,106)(39,172,182,122,107)(40,173,183,123,108)(41,174,184,124,97)(42,175,185,125,98)(43,176,186,126,99)(44,177,187,127,100)(45,178,188,128,101)(46,179,189,129,102)(47,180,190,130,103)(48,169,191,131,104)(145,220,233,203,165)(146,221,234,204,166)(147,222,235,193,167)(148,223,236,194,168)(149,224,237,195,157)(150,225,238,196,158)(151,226,239,197,159)(152,227,240,198,160)(153,228,229,199,161)(154,217,230,200,162)(155,218,231,201,163)(156,219,232,202,164), (1,117)(2,118)(3,119)(4,120)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,97)(48,98)(49,207)(50,208)(51,209)(52,210)(53,211)(54,212)(55,213)(56,214)(57,215)(58,216)(59,205)(60,206)(61,93)(62,94)(63,95)(64,96)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(121,177)(122,178)(123,179)(124,180)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(155,157)(156,158)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,228)(194,217)(195,218)(196,219)(197,220)(198,221)(199,222)(200,223)(201,224)(202,225)(203,226)(204,227)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (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), (1,154,7,148)(2,153,8,147)(3,152,9,146)(4,151,10,145)(5,150,11,156)(6,149,12,155)(13,239,19,233)(14,238,20,232)(15,237,21,231)(16,236,22,230)(17,235,23,229)(18,234,24,240)(25,48,31,42)(26,47,32,41)(27,46,33,40)(28,45,34,39)(29,44,35,38)(30,43,36,37)(49,220,55,226)(50,219,56,225)(51,218,57,224)(52,217,58,223)(53,228,59,222)(54,227,60,221)(61,169,67,175)(62,180,68,174)(63,179,69,173)(64,178,70,172)(65,177,71,171)(66,176,72,170)(73,184,79,190)(74,183,80,189)(75,182,81,188)(76,181,82,187)(77,192,83,186)(78,191,84,185)(85,121,91,127)(86,132,92,126)(87,131,93,125)(88,130,94,124)(89,129,95,123)(90,128,96,122)(97,142,103,136)(98,141,104,135)(99,140,105,134)(100,139,106,133)(101,138,107,144)(102,137,108,143)(109,164,115,158)(110,163,116,157)(111,162,117,168)(112,161,118,167)(113,160,119,166)(114,159,120,165)(193,205,199,211)(194,216,200,210)(195,215,201,209)(196,214,202,208)(197,213,203,207)(198,212,204,206)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,187)(14,188)(15,189)(16,190)(17,191)(18,192)(19,181)(20,182)(21,183)(22,184)(23,185)(24,186)(25,153)(26,154)(27,155)(28,156)(29,145)(30,146)(31,147)(32,148)(33,149)(34,150)(35,151)(36,152)(49,171)(50,172)(51,173)(52,174)(53,175)(54,176)(55,177)(56,178)(57,179)(58,180)(59,169)(60,170)(61,228)(62,217)(63,218)(64,219)(65,220)(66,221)(67,222)(68,223)(69,224)(70,225)(71,226)(72,227)(73,236)(74,237)(75,238)(76,239)(77,240)(78,229)(79,230)(80,231)(81,232)(82,233)(83,234)(84,235)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,193)(94,194)(95,195)(96,196)(97,111)(98,112)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,109)(108,110)(121,213)(122,214)(123,215)(124,216)(125,205)(126,206)(127,207)(128,208)(129,209)(130,210)(131,211)(132,212)(133,165)(134,166)(135,167)(136,168)(137,157)(138,158)(139,159)(140,160)(141,161)(142,162)(143,163)(144,164), (1,52,22,216,111)(2,53,23,205,112)(3,54,24,206,113)(4,55,13,207,114)(5,56,14,208,115)(6,57,15,209,116)(7,58,16,210,117)(8,59,17,211,118)(9,60,18,212,119)(10,49,19,213,120)(11,50,20,214,109)(12,51,21,215,110)(25,61,78,87,141)(26,62,79,88,142)(27,63,80,89,143)(28,64,81,90,144)(29,65,82,91,133)(30,66,83,92,134)(31,67,84,93,135)(32,68,73,94,136)(33,69,74,95,137)(34,70,75,96,138)(35,71,76,85,139)(36,72,77,86,140)(37,170,192,132,105)(38,171,181,121,106)(39,172,182,122,107)(40,173,183,123,108)(41,174,184,124,97)(42,175,185,125,98)(43,176,186,126,99)(44,177,187,127,100)(45,178,188,128,101)(46,179,189,129,102)(47,180,190,130,103)(48,169,191,131,104)(145,220,233,203,165)(146,221,234,204,166)(147,222,235,193,167)(148,223,236,194,168)(149,224,237,195,157)(150,225,238,196,158)(151,226,239,197,159)(152,227,240,198,160)(153,228,229,199,161)(154,217,230,200,162)(155,218,231,201,163)(156,219,232,202,164), (1,117)(2,118)(3,119)(4,120)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,97)(48,98)(49,207)(50,208)(51,209)(52,210)(53,211)(54,212)(55,213)(56,214)(57,215)(58,216)(59,205)(60,206)(61,93)(62,94)(63,95)(64,96)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(121,177)(122,178)(123,179)(124,180)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(155,157)(156,158)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,228)(194,217)(195,218)(196,219)(197,220)(198,221)(199,222)(200,223)(201,224)(202,225)(203,226)(204,227)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (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), (1,154,7,148)(2,153,8,147)(3,152,9,146)(4,151,10,145)(5,150,11,156)(6,149,12,155)(13,239,19,233)(14,238,20,232)(15,237,21,231)(16,236,22,230)(17,235,23,229)(18,234,24,240)(25,48,31,42)(26,47,32,41)(27,46,33,40)(28,45,34,39)(29,44,35,38)(30,43,36,37)(49,220,55,226)(50,219,56,225)(51,218,57,224)(52,217,58,223)(53,228,59,222)(54,227,60,221)(61,169,67,175)(62,180,68,174)(63,179,69,173)(64,178,70,172)(65,177,71,171)(66,176,72,170)(73,184,79,190)(74,183,80,189)(75,182,81,188)(76,181,82,187)(77,192,83,186)(78,191,84,185)(85,121,91,127)(86,132,92,126)(87,131,93,125)(88,130,94,124)(89,129,95,123)(90,128,96,122)(97,142,103,136)(98,141,104,135)(99,140,105,134)(100,139,106,133)(101,138,107,144)(102,137,108,143)(109,164,115,158)(110,163,116,157)(111,162,117,168)(112,161,118,167)(113,160,119,166)(114,159,120,165)(193,205,199,211)(194,216,200,210)(195,215,201,209)(196,214,202,208)(197,213,203,207)(198,212,204,206) );
G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,37),(10,38),(11,39),(12,40),(13,187),(14,188),(15,189),(16,190),(17,191),(18,192),(19,181),(20,182),(21,183),(22,184),(23,185),(24,186),(25,153),(26,154),(27,155),(28,156),(29,145),(30,146),(31,147),(32,148),(33,149),(34,150),(35,151),(36,152),(49,171),(50,172),(51,173),(52,174),(53,175),(54,176),(55,177),(56,178),(57,179),(58,180),(59,169),(60,170),(61,228),(62,217),(63,218),(64,219),(65,220),(66,221),(67,222),(68,223),(69,224),(70,225),(71,226),(72,227),(73,236),(74,237),(75,238),(76,239),(77,240),(78,229),(79,230),(80,231),(81,232),(82,233),(83,234),(84,235),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,193),(94,194),(95,195),(96,196),(97,111),(98,112),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(105,119),(106,120),(107,109),(108,110),(121,213),(122,214),(123,215),(124,216),(125,205),(126,206),(127,207),(128,208),(129,209),(130,210),(131,211),(132,212),(133,165),(134,166),(135,167),(136,168),(137,157),(138,158),(139,159),(140,160),(141,161),(142,162),(143,163),(144,164)], [(1,52,22,216,111),(2,53,23,205,112),(3,54,24,206,113),(4,55,13,207,114),(5,56,14,208,115),(6,57,15,209,116),(7,58,16,210,117),(8,59,17,211,118),(9,60,18,212,119),(10,49,19,213,120),(11,50,20,214,109),(12,51,21,215,110),(25,61,78,87,141),(26,62,79,88,142),(27,63,80,89,143),(28,64,81,90,144),(29,65,82,91,133),(30,66,83,92,134),(31,67,84,93,135),(32,68,73,94,136),(33,69,74,95,137),(34,70,75,96,138),(35,71,76,85,139),(36,72,77,86,140),(37,170,192,132,105),(38,171,181,121,106),(39,172,182,122,107),(40,173,183,123,108),(41,174,184,124,97),(42,175,185,125,98),(43,176,186,126,99),(44,177,187,127,100),(45,178,188,128,101),(46,179,189,129,102),(47,180,190,130,103),(48,169,191,131,104),(145,220,233,203,165),(146,221,234,204,166),(147,222,235,193,167),(148,223,236,194,168),(149,224,237,195,157),(150,225,238,196,158),(151,226,239,197,159),(152,227,240,198,160),(153,228,229,199,161),(154,217,230,200,162),(155,218,231,201,163),(156,219,232,202,164)], [(1,117),(2,118),(3,119),(4,120),(5,109),(6,110),(7,111),(8,112),(9,113),(10,114),(11,115),(12,116),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,133),(36,134),(37,99),(38,100),(39,101),(40,102),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,97),(48,98),(49,207),(50,208),(51,209),(52,210),(53,211),(54,212),(55,213),(56,214),(57,215),(58,216),(59,205),(60,206),(61,93),(62,94),(63,95),(64,96),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(121,177),(122,178),(123,179),(124,180),(125,169),(126,170),(127,171),(128,172),(129,173),(130,174),(131,175),(132,176),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(155,157),(156,158),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192),(193,228),(194,217),(195,218),(196,219),(197,220),(198,221),(199,222),(200,223),(201,224),(202,225),(203,226),(204,227),(229,235),(230,236),(231,237),(232,238),(233,239),(234,240)], [(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)], [(1,154,7,148),(2,153,8,147),(3,152,9,146),(4,151,10,145),(5,150,11,156),(6,149,12,155),(13,239,19,233),(14,238,20,232),(15,237,21,231),(16,236,22,230),(17,235,23,229),(18,234,24,240),(25,48,31,42),(26,47,32,41),(27,46,33,40),(28,45,34,39),(29,44,35,38),(30,43,36,37),(49,220,55,226),(50,219,56,225),(51,218,57,224),(52,217,58,223),(53,228,59,222),(54,227,60,221),(61,169,67,175),(62,180,68,174),(63,179,69,173),(64,178,70,172),(65,177,71,171),(66,176,72,170),(73,184,79,190),(74,183,80,189),(75,182,81,188),(76,181,82,187),(77,192,83,186),(78,191,84,185),(85,121,91,127),(86,132,92,126),(87,131,93,125),(88,130,94,124),(89,129,95,123),(90,128,96,122),(97,142,103,136),(98,141,104,135),(99,140,105,134),(100,139,106,133),(101,138,107,144),(102,137,108,143),(109,164,115,158),(110,163,116,157),(111,162,117,168),(112,161,118,167),(113,160,119,166),(114,159,120,165),(193,205,199,211),(194,216,200,210),(195,215,201,209),(196,214,202,208),(197,213,203,207),(198,212,204,206)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | + | + | - | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | Dic6 | S3×D5 | Q8×D5 | C2×S3×D5 | C2×S3×D5 | D5×Dic6 |
| kernel | C2×D5×Dic6 | D5×Dic6 | C2×D5×Dic3 | C2×C15⋊Q8 | D5×C2×C12 | C10×Dic6 | C2×Dic30 | C2×C4×D5 | C6×D5 | C2×Dic6 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | Dic6 | C2×Dic3 | C2×C12 | D10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 8 | 4 | 2 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C2×D5×Dic6 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 1 | 0 | 0 |
| 42 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 21 | 0 |
| 0 | 0 | 57 | 32 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 45 | 44 |
| 0 | 0 | 51 | 16 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[18,42,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,21,57,0,0,0,32],[60,0,0,0,0,60,0,0,0,0,45,51,0,0,44,16] >;
C2×D5×Dic6 in GAP, Magma, Sage, TeX
C_2\times D_5\times {\rm Dic}_6 % in TeX
G:=Group("C2xD5xDic6"); // GroupNames label
G:=SmallGroup(480,1073);
// by ID
G=gap.SmallGroup(480,1073);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^12=1,e^2=d^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations