direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D12⋊5D5, D12⋊26D10, C30.12C24, C60.136C23, Dic30⋊35C22, Dic15.7C23, (C4×D5)⋊14D6, (C10×D12)⋊7C2, (C2×D12)⋊15D5, C30⋊6(C4○D4), C10⋊3(C4○D12), C6⋊2(D4⋊2D5), (C2×C20).168D6, C15⋊D4⋊9C22, D6.4(C22×D5), C6.12(C23×D5), (C2×Dic30)⋊30C2, (C2×C12).310D10, (C5×D12)⋊22C22, (D5×C12)⋊16C22, (S3×C10).4C23, C10.12(S3×C23), (S3×Dic5)⋊6C22, (C22×D5).97D6, (C6×D5).39C23, (C2×C60).154C22, (C2×C30).231C23, C20.128(C22×S3), (C2×Dic5).219D6, (C22×S3).59D10, D10.41(C22×S3), C12.160(C22×D5), (C3×Dic5).41C23, Dic5.55(C22×S3), (C6×Dic5).228C22, (C2×Dic15).151C22, (C2×C4×D5)⋊3S3, (D5×C2×C12)⋊4C2, C15⋊6(C2×C4○D4), C5⋊3(C2×C4○D12), C4.86(C2×S3×D5), C3⋊2(C2×D4⋊2D5), (C2×S3×Dic5)⋊20C2, (C2×C15⋊D4)⋊18C2, C2.16(C22×S3×D5), (C2×C4).167(S3×D5), C22.100(C2×S3×D5), (S3×C2×C10).59C22, (D5×C2×C6).116C22, (C2×C6).241(C22×D5), (C2×C10).241(C22×S3), SmallGroup(480,1084)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1436 in 328 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×2], C10, C10 [×2], C10 [×4], Dic3 [×4], C12 [×2], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×8], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×C6, C5×S3 [×4], C3×D5 [×2], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C3×Dic5 [×2], Dic15 [×4], C60 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×4], S3×C10 [×4], C2×C30, C2×Dic10, C2×C4×D5, D4⋊2D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×C4○D12, S3×Dic5 [×8], C15⋊D4 [×8], D5×C12 [×4], C6×Dic5, C5×D12 [×4], Dic30 [×4], C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10 [×2], C2×D4⋊2D5, D12⋊5D5 [×8], C2×S3×Dic5 [×2], C2×C15⋊D4 [×2], D5×C2×C12, C10×D12, C2×Dic30, C2×D12⋊5D5
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, S3×D5, D4⋊2D5 [×2], C23×D5, C2×C4○D12, C2×S3×D5 [×3], C2×D4⋊2D5, D12⋊5D5 [×2], C22×S3×D5, C2×D12⋊5D5
Generators and relations
G = < a,b,c,d,e | a2=b12=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b6c, ede=d-1 >
►On 240 points
Generators in S240
(1 232)(2 233)(3 234)(4 235)(5 236)(6 237)(7 238)(8 239)(9 240)(10 229)(11 230)(12 231)(13 168)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 166)(24 167)(25 80)(26 81)(27 82)(28 83)(29 84)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 141)(57 142)(58 143)(59 144)(60 133)(61 195)(62 196)(63 197)(64 198)(65 199)(66 200)(67 201)(68 202)(69 203)(70 204)(71 193)(72 194)(97 118)(98 119)(99 120)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(127 190)(128 191)(129 192)(130 181)(131 182)(132 183)(145 208)(146 209)(147 210)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 205)(155 206)(156 207)(169 224)(170 225)(171 226)(172 227)(173 228)(174 217)(175 218)(176 219)(177 220)(178 221)(179 222)(180 223)
(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)
(1 178)(2 177)(3 176)(4 175)(5 174)(6 173)(7 172)(8 171)(9 170)(10 169)(11 180)(12 179)(13 121)(14 132)(15 131)(16 130)(17 129)(18 128)(19 127)(20 126)(21 125)(22 124)(23 123)(24 122)(25 212)(26 211)(27 210)(28 209)(29 208)(30 207)(31 206)(32 205)(33 216)(34 215)(35 214)(36 213)(37 133)(38 144)(39 143)(40 142)(41 141)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 111)(62 110)(63 109)(64 120)(65 119)(66 118)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 156)(74 155)(75 154)(76 153)(77 152)(78 151)(79 150)(80 149)(81 148)(82 147)(83 146)(84 145)(97 200)(98 199)(99 198)(100 197)(101 196)(102 195)(103 194)(104 193)(105 204)(106 203)(107 202)(108 201)(157 183)(158 182)(159 181)(160 192)(161 191)(162 190)(163 189)(164 188)(165 187)(166 186)(167 185)(168 184)(217 236)(218 235)(219 234)(220 233)(221 232)(222 231)(223 230)(224 229)(225 240)(226 239)(227 238)(228 237)
(1 134 98 167 32)(2 135 99 168 33)(3 136 100 157 34)(4 137 101 158 35)(5 138 102 159 36)(6 139 103 160 25)(7 140 104 161 26)(8 141 105 162 27)(9 142 106 163 28)(10 143 107 164 29)(11 144 108 165 30)(12 133 97 166 31)(13 76 233 50 120)(14 77 234 51 109)(15 78 235 52 110)(16 79 236 53 111)(17 80 237 54 112)(18 81 238 55 113)(19 82 239 56 114)(20 83 240 57 115)(21 84 229 58 116)(22 73 230 59 117)(23 74 231 60 118)(24 75 232 49 119)(37 200 186 206 179)(38 201 187 207 180)(39 202 188 208 169)(40 203 189 209 170)(41 204 190 210 171)(42 193 191 211 172)(43 194 192 212 173)(44 195 181 213 174)(45 196 182 214 175)(46 197 183 215 176)(47 198 184 216 177)(48 199 185 205 178)(61 130 150 217 92)(62 131 151 218 93)(63 132 152 219 94)(64 121 153 220 95)(65 122 154 221 96)(66 123 155 222 85)(67 124 156 223 86)(68 125 145 224 87)(69 126 146 225 88)(70 127 147 226 89)(71 128 148 227 90)(72 129 149 228 91)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(37 192)(38 181)(39 182)(40 183)(41 184)(42 185)(43 186)(44 187)(45 188)(46 189)(47 190)(48 191)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 230)(74 231)(75 232)(76 233)(77 234)(78 235)(79 236)(80 237)(81 238)(82 239)(83 240)(84 229)(85 129)(86 130)(87 131)(88 132)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(133 166)(134 167)(135 168)(136 157)(137 158)(138 159)(139 160)(140 161)(141 162)(142 163)(143 164)(144 165)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 224)(152 225)(153 226)(154 227)(155 228)(156 217)(169 214)(170 215)(171 216)(172 205)(173 206)(174 207)(175 208)(176 209)(177 210)(178 211)(179 212)(180 213)(193 199)(194 200)(195 201)(196 202)(197 203)(198 204)
G:=sub<Sym(240)| (1,232)(2,233)(3,234)(4,235)(5,236)(6,237)(7,238)(8,239)(9,240)(10,229)(11,230)(12,231)(13,168)(14,157)(15,158)(16,159)(17,160)(18,161)(19,162)(20,163)(21,164)(22,165)(23,166)(24,167)(25,80)(26,81)(27,82)(28,83)(29,84)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,141)(57,142)(58,143)(59,144)(60,133)(61,195)(62,196)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,193)(72,194)(97,118)(98,119)(99,120)(100,109)(101,110)(102,111)(103,112)(104,113)(105,114)(106,115)(107,116)(108,117)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,192)(130,181)(131,182)(132,183)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,205)(155,206)(156,207)(169,224)(170,225)(171,226)(172,227)(173,228)(174,217)(175,218)(176,219)(177,220)(178,221)(179,222)(180,223), (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), (1,178)(2,177)(3,176)(4,175)(5,174)(6,173)(7,172)(8,171)(9,170)(10,169)(11,180)(12,179)(13,121)(14,132)(15,131)(16,130)(17,129)(18,128)(19,127)(20,126)(21,125)(22,124)(23,123)(24,122)(25,212)(26,211)(27,210)(28,209)(29,208)(30,207)(31,206)(32,205)(33,216)(34,215)(35,214)(36,213)(37,133)(38,144)(39,143)(40,142)(41,141)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,111)(62,110)(63,109)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(81,148)(82,147)(83,146)(84,145)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,204)(106,203)(107,202)(108,201)(157,183)(158,182)(159,181)(160,192)(161,191)(162,190)(163,189)(164,188)(165,187)(166,186)(167,185)(168,184)(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)(225,240)(226,239)(227,238)(228,237), (1,134,98,167,32)(2,135,99,168,33)(3,136,100,157,34)(4,137,101,158,35)(5,138,102,159,36)(6,139,103,160,25)(7,140,104,161,26)(8,141,105,162,27)(9,142,106,163,28)(10,143,107,164,29)(11,144,108,165,30)(12,133,97,166,31)(13,76,233,50,120)(14,77,234,51,109)(15,78,235,52,110)(16,79,236,53,111)(17,80,237,54,112)(18,81,238,55,113)(19,82,239,56,114)(20,83,240,57,115)(21,84,229,58,116)(22,73,230,59,117)(23,74,231,60,118)(24,75,232,49,119)(37,200,186,206,179)(38,201,187,207,180)(39,202,188,208,169)(40,203,189,209,170)(41,204,190,210,171)(42,193,191,211,172)(43,194,192,212,173)(44,195,181,213,174)(45,196,182,214,175)(46,197,183,215,176)(47,198,184,216,177)(48,199,185,205,178)(61,130,150,217,92)(62,131,151,218,93)(63,132,152,219,94)(64,121,153,220,95)(65,122,154,221,96)(66,123,155,222,85)(67,124,156,223,86)(68,125,145,224,87)(69,126,146,225,88)(70,127,147,226,89)(71,128,148,227,90)(72,129,149,228,91), (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(37,192)(38,181)(39,182)(40,183)(41,184)(42,185)(43,186)(44,187)(45,188)(46,189)(47,190)(48,191)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,230)(74,231)(75,232)(76,233)(77,234)(78,235)(79,236)(80,237)(81,238)(82,239)(83,240)(84,229)(85,129)(86,130)(87,131)(88,132)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(133,166)(134,167)(135,168)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,225)(153,226)(154,227)(155,228)(156,217)(169,214)(170,215)(171,216)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,211)(179,212)(180,213)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)>;
G:=Group( (1,232)(2,233)(3,234)(4,235)(5,236)(6,237)(7,238)(8,239)(9,240)(10,229)(11,230)(12,231)(13,168)(14,157)(15,158)(16,159)(17,160)(18,161)(19,162)(20,163)(21,164)(22,165)(23,166)(24,167)(25,80)(26,81)(27,82)(28,83)(29,84)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,141)(57,142)(58,143)(59,144)(60,133)(61,195)(62,196)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,193)(72,194)(97,118)(98,119)(99,120)(100,109)(101,110)(102,111)(103,112)(104,113)(105,114)(106,115)(107,116)(108,117)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,192)(130,181)(131,182)(132,183)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,205)(155,206)(156,207)(169,224)(170,225)(171,226)(172,227)(173,228)(174,217)(175,218)(176,219)(177,220)(178,221)(179,222)(180,223), (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), (1,178)(2,177)(3,176)(4,175)(5,174)(6,173)(7,172)(8,171)(9,170)(10,169)(11,180)(12,179)(13,121)(14,132)(15,131)(16,130)(17,129)(18,128)(19,127)(20,126)(21,125)(22,124)(23,123)(24,122)(25,212)(26,211)(27,210)(28,209)(29,208)(30,207)(31,206)(32,205)(33,216)(34,215)(35,214)(36,213)(37,133)(38,144)(39,143)(40,142)(41,141)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,111)(62,110)(63,109)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(81,148)(82,147)(83,146)(84,145)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,204)(106,203)(107,202)(108,201)(157,183)(158,182)(159,181)(160,192)(161,191)(162,190)(163,189)(164,188)(165,187)(166,186)(167,185)(168,184)(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)(225,240)(226,239)(227,238)(228,237), (1,134,98,167,32)(2,135,99,168,33)(3,136,100,157,34)(4,137,101,158,35)(5,138,102,159,36)(6,139,103,160,25)(7,140,104,161,26)(8,141,105,162,27)(9,142,106,163,28)(10,143,107,164,29)(11,144,108,165,30)(12,133,97,166,31)(13,76,233,50,120)(14,77,234,51,109)(15,78,235,52,110)(16,79,236,53,111)(17,80,237,54,112)(18,81,238,55,113)(19,82,239,56,114)(20,83,240,57,115)(21,84,229,58,116)(22,73,230,59,117)(23,74,231,60,118)(24,75,232,49,119)(37,200,186,206,179)(38,201,187,207,180)(39,202,188,208,169)(40,203,189,209,170)(41,204,190,210,171)(42,193,191,211,172)(43,194,192,212,173)(44,195,181,213,174)(45,196,182,214,175)(46,197,183,215,176)(47,198,184,216,177)(48,199,185,205,178)(61,130,150,217,92)(62,131,151,218,93)(63,132,152,219,94)(64,121,153,220,95)(65,122,154,221,96)(66,123,155,222,85)(67,124,156,223,86)(68,125,145,224,87)(69,126,146,225,88)(70,127,147,226,89)(71,128,148,227,90)(72,129,149,228,91), (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(37,192)(38,181)(39,182)(40,183)(41,184)(42,185)(43,186)(44,187)(45,188)(46,189)(47,190)(48,191)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,230)(74,231)(75,232)(76,233)(77,234)(78,235)(79,236)(80,237)(81,238)(82,239)(83,240)(84,229)(85,129)(86,130)(87,131)(88,132)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(133,166)(134,167)(135,168)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,218)(146,219)(147,220)(148,221)(149,222)(150,223)(151,224)(152,225)(153,226)(154,227)(155,228)(156,217)(169,214)(170,215)(171,216)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,211)(179,212)(180,213)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204) );
G=PermutationGroup([(1,232),(2,233),(3,234),(4,235),(5,236),(6,237),(7,238),(8,239),(9,240),(10,229),(11,230),(12,231),(13,168),(14,157),(15,158),(16,159),(17,160),(18,161),(19,162),(20,163),(21,164),(22,165),(23,166),(24,167),(25,80),(26,81),(27,82),(28,83),(29,84),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,134),(50,135),(51,136),(52,137),(53,138),(54,139),(55,140),(56,141),(57,142),(58,143),(59,144),(60,133),(61,195),(62,196),(63,197),(64,198),(65,199),(66,200),(67,201),(68,202),(69,203),(70,204),(71,193),(72,194),(97,118),(98,119),(99,120),(100,109),(101,110),(102,111),(103,112),(104,113),(105,114),(106,115),(107,116),(108,117),(121,184),(122,185),(123,186),(124,187),(125,188),(126,189),(127,190),(128,191),(129,192),(130,181),(131,182),(132,183),(145,208),(146,209),(147,210),(148,211),(149,212),(150,213),(151,214),(152,215),(153,216),(154,205),(155,206),(156,207),(169,224),(170,225),(171,226),(172,227),(173,228),(174,217),(175,218),(176,219),(177,220),(178,221),(179,222),(180,223)], [(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)], [(1,178),(2,177),(3,176),(4,175),(5,174),(6,173),(7,172),(8,171),(9,170),(10,169),(11,180),(12,179),(13,121),(14,132),(15,131),(16,130),(17,129),(18,128),(19,127),(20,126),(21,125),(22,124),(23,123),(24,122),(25,212),(26,211),(27,210),(28,209),(29,208),(30,207),(31,206),(32,205),(33,216),(34,215),(35,214),(36,213),(37,133),(38,144),(39,143),(40,142),(41,141),(42,140),(43,139),(44,138),(45,137),(46,136),(47,135),(48,134),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(61,111),(62,110),(63,109),(64,120),(65,119),(66,118),(67,117),(68,116),(69,115),(70,114),(71,113),(72,112),(73,156),(74,155),(75,154),(76,153),(77,152),(78,151),(79,150),(80,149),(81,148),(82,147),(83,146),(84,145),(97,200),(98,199),(99,198),(100,197),(101,196),(102,195),(103,194),(104,193),(105,204),(106,203),(107,202),(108,201),(157,183),(158,182),(159,181),(160,192),(161,191),(162,190),(163,189),(164,188),(165,187),(166,186),(167,185),(168,184),(217,236),(218,235),(219,234),(220,233),(221,232),(222,231),(223,230),(224,229),(225,240),(226,239),(227,238),(228,237)], [(1,134,98,167,32),(2,135,99,168,33),(3,136,100,157,34),(4,137,101,158,35),(5,138,102,159,36),(6,139,103,160,25),(7,140,104,161,26),(8,141,105,162,27),(9,142,106,163,28),(10,143,107,164,29),(11,144,108,165,30),(12,133,97,166,31),(13,76,233,50,120),(14,77,234,51,109),(15,78,235,52,110),(16,79,236,53,111),(17,80,237,54,112),(18,81,238,55,113),(19,82,239,56,114),(20,83,240,57,115),(21,84,229,58,116),(22,73,230,59,117),(23,74,231,60,118),(24,75,232,49,119),(37,200,186,206,179),(38,201,187,207,180),(39,202,188,208,169),(40,203,189,209,170),(41,204,190,210,171),(42,193,191,211,172),(43,194,192,212,173),(44,195,181,213,174),(45,196,182,214,175),(46,197,183,215,176),(47,198,184,216,177),(48,199,185,205,178),(61,130,150,217,92),(62,131,151,218,93),(63,132,152,219,94),(64,121,153,220,95),(65,122,154,221,96),(66,123,155,222,85),(67,124,156,223,86),(68,125,145,224,87),(69,126,146,225,88),(70,127,147,226,89),(71,128,148,227,90),(72,129,149,228,91)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,49),(37,192),(38,181),(39,182),(40,183),(41,184),(42,185),(43,186),(44,187),(45,188),(46,189),(47,190),(48,191),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,230),(74,231),(75,232),(76,233),(77,234),(78,235),(79,236),(80,237),(81,238),(82,239),(83,240),(84,229),(85,129),(86,130),(87,131),(88,132),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128),(133,166),(134,167),(135,168),(136,157),(137,158),(138,159),(139,160),(140,161),(141,162),(142,163),(143,164),(144,165),(145,218),(146,219),(147,220),(148,221),(149,222),(150,223),(151,224),(152,225),(153,226),(154,227),(155,228),(156,217),(169,214),(170,215),(171,216),(172,205),(173,206),(174,207),(175,208),(176,209),(177,210),(178,211),(179,212),(180,213),(193,199),(194,200),(195,201),(196,202),(197,203),(198,204)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 29 | 0 | 0 | 0 |
| 51 | 40 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 46 | 47 | 0 | 0 |
| 16 | 15 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 17 |
| 1 | 0 | 0 | 0 |
| 24 | 60 | 0 | 0 |
| 0 | 0 | 44 | 60 |
| 0 | 0 | 44 | 17 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[29,51,0,0,0,40,0,0,0,0,60,0,0,0,0,60],[46,16,0,0,47,15,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,17],[1,24,0,0,0,60,0,0,0,0,44,44,0,0,60,17] >;
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | D5 | D6 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | S3×D5 | D4⋊2D5 | C2×S3×D5 | C2×S3×D5 | D12⋊5D5 |
| kernel | C2×D12⋊5D5 | D12⋊5D5 | C2×S3×Dic5 | C2×C15⋊D4 | D5×C2×C12 | C10×D12 | C2×Dic30 | C2×C4×D5 | C2×D12 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C30 | D12 | C2×C12 | C22×S3 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 4 | 8 | 2 | 4 | 8 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_{12}\rtimes_5D_5 % in TeX
G:=Group("C2xD12:5D5"); // GroupNames label
G:=SmallGroup(480,1084);
// by ID
G=gap.SmallGroup(480,1084);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^5=e^2=1,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,e*c*e=b^6*c,e*d*e=d^-1>;
// generators/relations
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