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, C20.121(C22×S3), (C2×C60).153C22, (C2×Dic5).197D6, (C5×Dic6)⋊20C22, (D5×C12).96C22, D10.52(C22×S3), (C22×D5).111D6, C12.158(C22×D5), Dic3.1(C22×D5), (C5×Dic3).1C23, (D5×Dic3).9C22, (C2×Dic3).128D10, Dic5.41(C22×S3), (C3×Dic5).39C23, (C6×Dic5).226C22, (C2×Dic15).147C22, (C10×Dic3).126C22, 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
Subgroups: 1308 in 312 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×10], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×17], Q8 [×16], C23, D5 [×4], C10, C10 [×2], Dic3 [×4], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C15, C22×C4 [×3], C2×Q8 [×12], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], C2×C10, Dic6 [×4], Dic6 [×12], C2×Dic3 [×2], C2×Dic3 [×10], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×4], C30, C30 [×2], C22×Q8, Dic10 [×12], C4×D5 [×4], C4×D5 [×8], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C2×Dic6, C2×Dic6 [×11], C22×Dic3 [×2], C22×C12, C5×Dic3 [×4], C3×Dic5 [×2], Dic15 [×4], C60 [×2], C6×D5 [×6], C2×C30, C2×Dic10 [×3], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×8], Q8×C10, C22×Dic6, D5×Dic3 [×8], C15⋊Q8 [×8], D5×C12 [×4], C6×Dic5, C5×Dic6 [×4], C10×Dic3 [×2], Dic30 [×4], C2×Dic15 [×2], C2×C60, D5×C2×C6, C2×Q8×D5, D5×Dic6 [×8], C2×D5×Dic3 [×2], C2×C15⋊Q8 [×2], D5×C2×C12, C10×Dic6, C2×Dic30, C2×D5×Dic6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D5, D6 [×7], C2×Q8 [×6], C24, D10 [×7], Dic6 [×4], C22×S3 [×7], C22×Q8, C22×D5 [×7], C2×Dic6 [×6], S3×C23, S3×D5, Q8×D5 [×2], C23×D5, C22×Dic6, C2×S3×D5 [×3], C2×Q8×D5, D5×Dic6 [×2], C22×S3×D5, C2×D5×Dic6
Generators and relations
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 >
►On 240 points
Generators in S240
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 207)(14 208)(15 209)(16 210)(17 211)(18 212)(19 213)(20 214)(21 215)(22 216)(23 205)(24 206)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 121)(35 122)(36 123)(37 155)(38 156)(39 145)(40 146)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 181)(50 182)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(61 174)(62 175)(63 176)(64 177)(65 178)(66 179)(67 180)(68 169)(69 170)(70 171)(71 172)(72 173)(73 198)(74 199)(75 200)(76 201)(77 202)(78 203)(79 204)(80 193)(81 194)(82 195)(83 196)(84 197)(85 136)(86 137)(87 138)(88 139)(89 140)(90 141)(91 142)(92 143)(93 144)(94 133)(95 134)(96 135)(97 223)(98 224)(99 225)(100 226)(101 227)(102 228)(103 217)(104 218)(105 219)(106 220)(107 221)(108 222)(109 233)(110 234)(111 235)(112 236)(113 237)(114 238)(115 239)(116 240)(117 229)(118 230)(119 231)(120 232)
(1 156 68 195 58)(2 145 69 196 59)(3 146 70 197 60)(4 147 71 198 49)(5 148 72 199 50)(6 149 61 200 51)(7 150 62 201 52)(8 151 63 202 53)(9 152 64 203 54)(10 153 65 204 55)(11 154 66 193 56)(12 155 67 194 57)(13 110 227 28 144)(14 111 228 29 133)(15 112 217 30 134)(16 113 218 31 135)(17 114 219 32 136)(18 115 220 33 137)(19 116 221 34 138)(20 117 222 35 139)(21 118 223 36 140)(22 119 224 25 141)(23 120 225 26 142)(24 109 226 27 143)(37 180 81 189 168)(38 169 82 190 157)(39 170 83 191 158)(40 171 84 192 159)(41 172 73 181 160)(42 173 74 182 161)(43 174 75 183 162)(44 175 76 184 163)(45 176 77 185 164)(46 177 78 186 165)(47 178 79 187 166)(48 179 80 188 167)(85 211 238 105 131)(86 212 239 106 132)(87 213 240 107 121)(88 214 229 108 122)(89 215 230 97 123)(90 216 231 98 124)(91 205 232 99 125)(92 206 233 100 126)(93 207 234 101 127)(94 208 235 102 128)(95 209 236 103 129)(96 210 237 104 130)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 49)(11 50)(12 51)(13 138)(14 139)(15 140)(16 141)(17 142)(18 143)(19 144)(20 133)(21 134)(22 135)(23 136)(24 137)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 109)(34 110)(35 111)(36 112)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 73)(48 74)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(121 234)(122 235)(123 236)(124 237)(125 238)(126 239)(127 240)(128 229)(129 230)(130 231)(131 232)(132 233)(145 202)(146 203)(147 204)(148 193)(149 194)(150 195)(151 196)(152 197)(153 198)(154 199)(155 200)(156 201)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 190)(164 191)(165 192)(166 181)(167 182)(168 183)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(217 223)(218 224)(219 225)(220 226)(221 227)(222 228)
(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 14 7 20)(2 13 8 19)(3 24 9 18)(4 23 10 17)(5 22 11 16)(6 21 12 15)(25 193 31 199)(26 204 32 198)(27 203 33 197)(28 202 34 196)(29 201 35 195)(30 200 36 194)(37 236 43 230)(38 235 44 229)(39 234 45 240)(40 233 46 239)(41 232 47 238)(42 231 48 237)(49 142 55 136)(50 141 56 135)(51 140 57 134)(52 139 58 133)(53 138 59 144)(54 137 60 143)(61 223 67 217)(62 222 68 228)(63 221 69 227)(64 220 70 226)(65 219 71 225)(66 218 72 224)(73 125 79 131)(74 124 80 130)(75 123 81 129)(76 122 82 128)(77 121 83 127)(78 132 84 126)(85 181 91 187)(86 192 92 186)(87 191 93 185)(88 190 94 184)(89 189 95 183)(90 188 96 182)(97 180 103 174)(98 179 104 173)(99 178 105 172)(100 177 106 171)(101 176 107 170)(102 175 108 169)(109 152 115 146)(110 151 116 145)(111 150 117 156)(112 149 118 155)(113 148 119 154)(114 147 120 153)(157 208 163 214)(158 207 164 213)(159 206 165 212)(160 205 166 211)(161 216 167 210)(162 215 168 209)
G:=sub<Sym(240)| (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,207)(14,208)(15,209)(16,210)(17,211)(18,212)(19,213)(20,214)(21,215)(22,216)(23,205)(24,206)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,121)(35,122)(36,123)(37,155)(38,156)(39,145)(40,146)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,181)(50,182)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(61,174)(62,175)(63,176)(64,177)(65,178)(66,179)(67,180)(68,169)(69,170)(70,171)(71,172)(72,173)(73,198)(74,199)(75,200)(76,201)(77,202)(78,203)(79,204)(80,193)(81,194)(82,195)(83,196)(84,197)(85,136)(86,137)(87,138)(88,139)(89,140)(90,141)(91,142)(92,143)(93,144)(94,133)(95,134)(96,135)(97,223)(98,224)(99,225)(100,226)(101,227)(102,228)(103,217)(104,218)(105,219)(106,220)(107,221)(108,222)(109,233)(110,234)(111,235)(112,236)(113,237)(114,238)(115,239)(116,240)(117,229)(118,230)(119,231)(120,232), (1,156,68,195,58)(2,145,69,196,59)(3,146,70,197,60)(4,147,71,198,49)(5,148,72,199,50)(6,149,61,200,51)(7,150,62,201,52)(8,151,63,202,53)(9,152,64,203,54)(10,153,65,204,55)(11,154,66,193,56)(12,155,67,194,57)(13,110,227,28,144)(14,111,228,29,133)(15,112,217,30,134)(16,113,218,31,135)(17,114,219,32,136)(18,115,220,33,137)(19,116,221,34,138)(20,117,222,35,139)(21,118,223,36,140)(22,119,224,25,141)(23,120,225,26,142)(24,109,226,27,143)(37,180,81,189,168)(38,169,82,190,157)(39,170,83,191,158)(40,171,84,192,159)(41,172,73,181,160)(42,173,74,182,161)(43,174,75,183,162)(44,175,76,184,163)(45,176,77,185,164)(46,177,78,186,165)(47,178,79,187,166)(48,179,80,188,167)(85,211,238,105,131)(86,212,239,106,132)(87,213,240,107,121)(88,214,229,108,122)(89,215,230,97,123)(90,216,231,98,124)(91,205,232,99,125)(92,206,233,100,126)(93,207,234,101,127)(94,208,235,102,128)(95,209,236,103,129)(96,210,237,104,130), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,138)(14,139)(15,140)(16,141)(17,142)(18,143)(19,144)(20,133)(21,134)(22,135)(23,136)(24,137)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,109)(34,110)(35,111)(36,112)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,73)(48,74)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,229)(129,230)(130,231)(131,232)(132,233)(145,202)(146,203)(147,204)(148,193)(149,194)(150,195)(151,196)(152,197)(153,198)(154,199)(155,200)(156,201)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,181)(167,182)(168,183)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228), (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), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,193,31,199)(26,204,32,198)(27,203,33,197)(28,202,34,196)(29,201,35,195)(30,200,36,194)(37,236,43,230)(38,235,44,229)(39,234,45,240)(40,233,46,239)(41,232,47,238)(42,231,48,237)(49,142,55,136)(50,141,56,135)(51,140,57,134)(52,139,58,133)(53,138,59,144)(54,137,60,143)(61,223,67,217)(62,222,68,228)(63,221,69,227)(64,220,70,226)(65,219,71,225)(66,218,72,224)(73,125,79,131)(74,124,80,130)(75,123,81,129)(76,122,82,128)(77,121,83,127)(78,132,84,126)(85,181,91,187)(86,192,92,186)(87,191,93,185)(88,190,94,184)(89,189,95,183)(90,188,96,182)(97,180,103,174)(98,179,104,173)(99,178,105,172)(100,177,106,171)(101,176,107,170)(102,175,108,169)(109,152,115,146)(110,151,116,145)(111,150,117,156)(112,149,118,155)(113,148,119,154)(114,147,120,153)(157,208,163,214)(158,207,164,213)(159,206,165,212)(160,205,166,211)(161,216,167,210)(162,215,168,209)>;
G:=Group( (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,207)(14,208)(15,209)(16,210)(17,211)(18,212)(19,213)(20,214)(21,215)(22,216)(23,205)(24,206)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,121)(35,122)(36,123)(37,155)(38,156)(39,145)(40,146)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,181)(50,182)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(61,174)(62,175)(63,176)(64,177)(65,178)(66,179)(67,180)(68,169)(69,170)(70,171)(71,172)(72,173)(73,198)(74,199)(75,200)(76,201)(77,202)(78,203)(79,204)(80,193)(81,194)(82,195)(83,196)(84,197)(85,136)(86,137)(87,138)(88,139)(89,140)(90,141)(91,142)(92,143)(93,144)(94,133)(95,134)(96,135)(97,223)(98,224)(99,225)(100,226)(101,227)(102,228)(103,217)(104,218)(105,219)(106,220)(107,221)(108,222)(109,233)(110,234)(111,235)(112,236)(113,237)(114,238)(115,239)(116,240)(117,229)(118,230)(119,231)(120,232), (1,156,68,195,58)(2,145,69,196,59)(3,146,70,197,60)(4,147,71,198,49)(5,148,72,199,50)(6,149,61,200,51)(7,150,62,201,52)(8,151,63,202,53)(9,152,64,203,54)(10,153,65,204,55)(11,154,66,193,56)(12,155,67,194,57)(13,110,227,28,144)(14,111,228,29,133)(15,112,217,30,134)(16,113,218,31,135)(17,114,219,32,136)(18,115,220,33,137)(19,116,221,34,138)(20,117,222,35,139)(21,118,223,36,140)(22,119,224,25,141)(23,120,225,26,142)(24,109,226,27,143)(37,180,81,189,168)(38,169,82,190,157)(39,170,83,191,158)(40,171,84,192,159)(41,172,73,181,160)(42,173,74,182,161)(43,174,75,183,162)(44,175,76,184,163)(45,176,77,185,164)(46,177,78,186,165)(47,178,79,187,166)(48,179,80,188,167)(85,211,238,105,131)(86,212,239,106,132)(87,213,240,107,121)(88,214,229,108,122)(89,215,230,97,123)(90,216,231,98,124)(91,205,232,99,125)(92,206,233,100,126)(93,207,234,101,127)(94,208,235,102,128)(95,209,236,103,129)(96,210,237,104,130), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,138)(14,139)(15,140)(16,141)(17,142)(18,143)(19,144)(20,133)(21,134)(22,135)(23,136)(24,137)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,109)(34,110)(35,111)(36,112)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,73)(48,74)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,229)(129,230)(130,231)(131,232)(132,233)(145,202)(146,203)(147,204)(148,193)(149,194)(150,195)(151,196)(152,197)(153,198)(154,199)(155,200)(156,201)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,181)(167,182)(168,183)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228), (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), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,193,31,199)(26,204,32,198)(27,203,33,197)(28,202,34,196)(29,201,35,195)(30,200,36,194)(37,236,43,230)(38,235,44,229)(39,234,45,240)(40,233,46,239)(41,232,47,238)(42,231,48,237)(49,142,55,136)(50,141,56,135)(51,140,57,134)(52,139,58,133)(53,138,59,144)(54,137,60,143)(61,223,67,217)(62,222,68,228)(63,221,69,227)(64,220,70,226)(65,219,71,225)(66,218,72,224)(73,125,79,131)(74,124,80,130)(75,123,81,129)(76,122,82,128)(77,121,83,127)(78,132,84,126)(85,181,91,187)(86,192,92,186)(87,191,93,185)(88,190,94,184)(89,189,95,183)(90,188,96,182)(97,180,103,174)(98,179,104,173)(99,178,105,172)(100,177,106,171)(101,176,107,170)(102,175,108,169)(109,152,115,146)(110,151,116,145)(111,150,117,156)(112,149,118,155)(113,148,119,154)(114,147,120,153)(157,208,163,214)(158,207,164,213)(159,206,165,212)(160,205,166,211)(161,216,167,210)(162,215,168,209) );
G=PermutationGroup([(1,157),(2,158),(3,159),(4,160),(5,161),(6,162),(7,163),(8,164),(9,165),(10,166),(11,167),(12,168),(13,207),(14,208),(15,209),(16,210),(17,211),(18,212),(19,213),(20,214),(21,215),(22,216),(23,205),(24,206),(25,124),(26,125),(27,126),(28,127),(29,128),(30,129),(31,130),(32,131),(33,132),(34,121),(35,122),(36,123),(37,155),(38,156),(39,145),(40,146),(41,147),(42,148),(43,149),(44,150),(45,151),(46,152),(47,153),(48,154),(49,181),(50,182),(51,183),(52,184),(53,185),(54,186),(55,187),(56,188),(57,189),(58,190),(59,191),(60,192),(61,174),(62,175),(63,176),(64,177),(65,178),(66,179),(67,180),(68,169),(69,170),(70,171),(71,172),(72,173),(73,198),(74,199),(75,200),(76,201),(77,202),(78,203),(79,204),(80,193),(81,194),(82,195),(83,196),(84,197),(85,136),(86,137),(87,138),(88,139),(89,140),(90,141),(91,142),(92,143),(93,144),(94,133),(95,134),(96,135),(97,223),(98,224),(99,225),(100,226),(101,227),(102,228),(103,217),(104,218),(105,219),(106,220),(107,221),(108,222),(109,233),(110,234),(111,235),(112,236),(113,237),(114,238),(115,239),(116,240),(117,229),(118,230),(119,231),(120,232)], [(1,156,68,195,58),(2,145,69,196,59),(3,146,70,197,60),(4,147,71,198,49),(5,148,72,199,50),(6,149,61,200,51),(7,150,62,201,52),(8,151,63,202,53),(9,152,64,203,54),(10,153,65,204,55),(11,154,66,193,56),(12,155,67,194,57),(13,110,227,28,144),(14,111,228,29,133),(15,112,217,30,134),(16,113,218,31,135),(17,114,219,32,136),(18,115,220,33,137),(19,116,221,34,138),(20,117,222,35,139),(21,118,223,36,140),(22,119,224,25,141),(23,120,225,26,142),(24,109,226,27,143),(37,180,81,189,168),(38,169,82,190,157),(39,170,83,191,158),(40,171,84,192,159),(41,172,73,181,160),(42,173,74,182,161),(43,174,75,183,162),(44,175,76,184,163),(45,176,77,185,164),(46,177,78,186,165),(47,178,79,187,166),(48,179,80,188,167),(85,211,238,105,131),(86,212,239,106,132),(87,213,240,107,121),(88,214,229,108,122),(89,215,230,97,123),(90,216,231,98,124),(91,205,232,99,125),(92,206,233,100,126),(93,207,234,101,127),(94,208,235,102,128),(95,209,236,103,129),(96,210,237,104,130)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,49),(11,50),(12,51),(13,138),(14,139),(15,140),(16,141),(17,142),(18,143),(19,144),(20,133),(21,134),(22,135),(23,136),(24,137),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,109),(34,110),(35,111),(36,112),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(46,84),(47,73),(48,74),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108),(121,234),(122,235),(123,236),(124,237),(125,238),(126,239),(127,240),(128,229),(129,230),(130,231),(131,232),(132,233),(145,202),(146,203),(147,204),(148,193),(149,194),(150,195),(151,196),(152,197),(153,198),(154,199),(155,200),(156,201),(157,184),(158,185),(159,186),(160,187),(161,188),(162,189),(163,190),(164,191),(165,192),(166,181),(167,182),(168,183),(169,175),(170,176),(171,177),(172,178),(173,179),(174,180),(217,223),(218,224),(219,225),(220,226),(221,227),(222,228)], [(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)], [(1,14,7,20),(2,13,8,19),(3,24,9,18),(4,23,10,17),(5,22,11,16),(6,21,12,15),(25,193,31,199),(26,204,32,198),(27,203,33,197),(28,202,34,196),(29,201,35,195),(30,200,36,194),(37,236,43,230),(38,235,44,229),(39,234,45,240),(40,233,46,239),(41,232,47,238),(42,231,48,237),(49,142,55,136),(50,141,56,135),(51,140,57,134),(52,139,58,133),(53,138,59,144),(54,137,60,143),(61,223,67,217),(62,222,68,228),(63,221,69,227),(64,220,70,226),(65,219,71,225),(66,218,72,224),(73,125,79,131),(74,124,80,130),(75,123,81,129),(76,122,82,128),(77,121,83,127),(78,132,84,126),(85,181,91,187),(86,192,92,186),(87,191,93,185),(88,190,94,184),(89,189,95,183),(90,188,96,182),(97,180,103,174),(98,179,104,173),(99,178,105,172),(100,177,106,171),(101,176,107,170),(102,175,108,169),(109,152,115,146),(110,151,116,145),(111,150,117,156),(112,149,118,155),(113,148,119,154),(114,147,120,153),(157,208,163,214),(158,207,164,213),(159,206,165,212),(160,205,166,211),(161,216,167,210),(162,215,168,209)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
C_2\times D_5\times 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
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