direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D12⋊D5, D12⋊22D10, C30.7C24, Dic10⋊20D6, C60.160C23, D30.33C23, Dic15.35C23, (C2×D12)⋊12D5, C30⋊3(C4○D4), (C10×D12)⋊12C2, C6⋊1(D4⋊2D5), C6.7(C23×D5), (C2×C20).165D6, C5⋊D12⋊8C22, C10.7(S3×C23), D6.2(C22×D5), C10⋊2(Q8⋊3S3), (C2×Dic10)⋊12S3, (C6×Dic10)⋊12C2, (C2×C12).164D10, (C4×D15)⋊22C22, (C5×D12)⋊29C22, (S3×C10).2C23, (S3×Dic5)⋊5C22, (C2×C60).208C22, C20.125(C22×S3), (C2×C30).226C23, (C2×Dic5).136D6, (C22×S3).58D10, C12.125(C22×D5), Dic5.4(C22×S3), (C3×Dic5).4C23, (C3×Dic10)⋊26C22, (C6×Dic5).127C22, (C2×Dic15).232C22, (C22×D15).117C22, C15⋊3(C2×C4○D4), (C2×C4×D15)⋊25C2, C3⋊1(C2×D4⋊2D5), C5⋊2(C2×Q8⋊3S3), C4.132(C2×S3×D5), (C2×S3×Dic5)⋊19C2, (C2×C5⋊D12)⋊17C2, C2.11(C22×S3×D5), C22.96(C2×S3×D5), (C2×C4).218(S3×D5), (S3×C2×C10).58C22, (C2×C6).236(C22×D5), (C2×C10).237(C22×S3), SmallGroup(480,1079)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D12⋊D5
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, ebe=b5, cd=dc, ece=b10c, ede=d-1 >
Subgroups: 1532 in 328 conjugacy classes, 116 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C5×S3, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, S3×C2×C4, C2×D12, C2×D12, Q8⋊3S3, C6×Q8, C3×Dic5, Dic15, C60, S3×C10, S3×C10, D30, D30, C2×C30, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C2×Q8⋊3S3, S3×Dic5, C5⋊D12, C3×Dic10, C6×Dic5, C5×D12, C4×D15, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C2×D4⋊2D5, D12⋊D5, C2×S3×Dic5, C2×C5⋊D12, C6×Dic10, C10×D12, C2×C4×D15, C2×D12⋊D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, Q8⋊3S3, S3×C23, S3×D5, D4⋊2D5, C23×D5, C2×Q8⋊3S3, C2×S3×D5, C2×D4⋊2D5, D12⋊D5, C22×S3×D5, C2×D12⋊D5
►On 240 points
Generators in S240
(1 221)(2 222)(3 223)(4 224)(5 225)(6 226)(7 227)(8 228)(9 217)(10 218)(11 219)(12 220)(13 229)(14 230)(15 231)(16 232)(17 233)(18 234)(19 235)(20 236)(21 237)(22 238)(23 239)(24 240)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 109)(33 110)(34 111)(35 112)(36 113)(37 182)(38 183)(39 184)(40 185)(41 186)(42 187)(43 188)(44 189)(45 190)(46 191)(47 192)(48 181)(49 130)(50 131)(51 132)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 127)(59 128)(60 129)(61 159)(62 160)(63 161)(64 162)(65 163)(66 164)(67 165)(68 166)(69 167)(70 168)(71 157)(72 158)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 85)(82 86)(83 87)(84 88)(97 140)(98 141)(99 142)(100 143)(101 144)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(145 213)(146 214)(147 215)(148 216)(149 205)(150 206)(151 207)(152 208)(153 209)(154 210)(155 211)(156 212)(169 196)(170 197)(171 198)(172 199)(173 200)(174 201)(175 202)(176 203)(177 204)(178 193)(179 194)(180 195)
(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)
(1 220)(2 219)(3 218)(4 217)(5 228)(6 227)(7 226)(8 225)(9 224)(10 223)(11 222)(12 221)(13 238)(14 237)(15 236)(16 235)(17 234)(18 233)(19 232)(20 231)(21 230)(22 229)(23 240)(24 239)(25 111)(26 110)(27 109)(28 120)(29 119)(30 118)(31 117)(32 116)(33 115)(34 114)(35 113)(36 112)(37 187)(38 186)(39 185)(40 184)(41 183)(42 182)(43 181)(44 192)(45 191)(46 190)(47 189)(48 188)(49 127)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 132)(57 131)(58 130)(59 129)(60 128)(61 160)(62 159)(63 158)(64 157)(65 168)(66 167)(67 166)(68 165)(69 164)(70 163)(71 162)(72 161)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 96)(82 95)(83 94)(84 93)(97 143)(98 142)(99 141)(100 140)(101 139)(102 138)(103 137)(104 136)(105 135)(106 134)(107 133)(108 144)(145 216)(146 215)(147 214)(148 213)(149 212)(150 211)(151 210)(152 209)(153 208)(154 207)(155 206)(156 205)(169 193)(170 204)(171 203)(172 202)(173 201)(174 200)(175 199)(176 198)(177 197)(178 196)(179 195)(180 194)
(1 123 153 142 191)(2 124 154 143 192)(3 125 155 144 181)(4 126 156 133 182)(5 127 145 134 183)(6 128 146 135 184)(7 129 147 136 185)(8 130 148 137 186)(9 131 149 138 187)(10 132 150 139 188)(11 121 151 140 189)(12 122 152 141 190)(13 196 161 114 76)(14 197 162 115 77)(15 198 163 116 78)(16 199 164 117 79)(17 200 165 118 80)(18 201 166 119 81)(19 202 167 120 82)(20 203 168 109 83)(21 204 157 110 84)(22 193 158 111 73)(23 194 159 112 74)(24 195 160 113 75)(25 92 229 169 63)(26 93 230 170 64)(27 94 231 171 65)(28 95 232 172 66)(29 96 233 173 67)(30 85 234 174 68)(31 86 235 175 69)(32 87 236 176 70)(33 88 237 177 71)(34 89 238 178 72)(35 90 239 179 61)(36 91 240 180 62)(37 224 57 212 102)(38 225 58 213 103)(39 226 59 214 104)(40 227 60 215 105)(41 228 49 216 106)(42 217 50 205 107)(43 218 51 206 108)(44 219 52 207 97)(45 220 53 208 98)(46 221 54 209 99)(47 222 55 210 100)(48 223 56 211 101)
(1 231)(2 236)(3 229)(4 234)(5 239)(6 232)(7 237)(8 230)(9 235)(10 240)(11 233)(12 238)(13 223)(14 228)(15 221)(16 226)(17 219)(18 224)(19 217)(20 222)(21 227)(22 220)(23 225)(24 218)(25 155)(26 148)(27 153)(28 146)(29 151)(30 156)(31 149)(32 154)(33 147)(34 152)(35 145)(36 150)(37 201)(38 194)(39 199)(40 204)(41 197)(42 202)(43 195)(44 200)(45 193)(46 198)(47 203)(48 196)(49 77)(50 82)(51 75)(52 80)(53 73)(54 78)(55 83)(56 76)(57 81)(58 74)(59 79)(60 84)(61 134)(62 139)(63 144)(64 137)(65 142)(66 135)(67 140)(68 133)(69 138)(70 143)(71 136)(72 141)(85 126)(86 131)(87 124)(88 129)(89 122)(90 127)(91 132)(92 125)(93 130)(94 123)(95 128)(96 121)(97 165)(98 158)(99 163)(100 168)(101 161)(102 166)(103 159)(104 164)(105 157)(106 162)(107 167)(108 160)(109 210)(110 215)(111 208)(112 213)(113 206)(114 211)(115 216)(116 209)(117 214)(118 207)(119 212)(120 205)(169 181)(170 186)(171 191)(172 184)(173 189)(174 182)(175 187)(176 192)(177 185)(178 190)(179 183)(180 188)
G:=sub<Sym(240)| (1,221)(2,222)(3,223)(4,224)(5,225)(6,226)(7,227)(8,228)(9,217)(10,218)(11,219)(12,220)(13,229)(14,230)(15,231)(16,232)(17,233)(18,234)(19,235)(20,236)(21,237)(22,238)(23,239)(24,240)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,109)(33,110)(34,111)(35,112)(36,113)(37,182)(38,183)(39,184)(40,185)(41,186)(42,187)(43,188)(44,189)(45,190)(46,191)(47,192)(48,181)(49,130)(50,131)(51,132)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,127)(59,128)(60,129)(61,159)(62,160)(63,161)(64,162)(65,163)(66,164)(67,165)(68,166)(69,167)(70,168)(71,157)(72,158)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,85)(82,86)(83,87)(84,88)(97,140)(98,141)(99,142)(100,143)(101,144)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(145,213)(146,214)(147,215)(148,216)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,211)(156,212)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)(176,203)(177,204)(178,193)(179,194)(180,195), (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), (1,220)(2,219)(3,218)(4,217)(5,228)(6,227)(7,226)(8,225)(9,224)(10,223)(11,222)(12,221)(13,238)(14,237)(15,236)(16,235)(17,234)(18,233)(19,232)(20,231)(21,230)(22,229)(23,240)(24,239)(25,111)(26,110)(27,109)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,187)(38,186)(39,185)(40,184)(41,183)(42,182)(43,181)(44,192)(45,191)(46,190)(47,189)(48,188)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,132)(57,131)(58,130)(59,129)(60,128)(61,160)(62,159)(63,158)(64,157)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,96)(82,95)(83,94)(84,93)(97,143)(98,142)(99,141)(100,140)(101,139)(102,138)(103,137)(104,136)(105,135)(106,134)(107,133)(108,144)(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)(152,209)(153,208)(154,207)(155,206)(156,205)(169,193)(170,204)(171,203)(172,202)(173,201)(174,200)(175,199)(176,198)(177,197)(178,196)(179,195)(180,194), (1,123,153,142,191)(2,124,154,143,192)(3,125,155,144,181)(4,126,156,133,182)(5,127,145,134,183)(6,128,146,135,184)(7,129,147,136,185)(8,130,148,137,186)(9,131,149,138,187)(10,132,150,139,188)(11,121,151,140,189)(12,122,152,141,190)(13,196,161,114,76)(14,197,162,115,77)(15,198,163,116,78)(16,199,164,117,79)(17,200,165,118,80)(18,201,166,119,81)(19,202,167,120,82)(20,203,168,109,83)(21,204,157,110,84)(22,193,158,111,73)(23,194,159,112,74)(24,195,160,113,75)(25,92,229,169,63)(26,93,230,170,64)(27,94,231,171,65)(28,95,232,172,66)(29,96,233,173,67)(30,85,234,174,68)(31,86,235,175,69)(32,87,236,176,70)(33,88,237,177,71)(34,89,238,178,72)(35,90,239,179,61)(36,91,240,180,62)(37,224,57,212,102)(38,225,58,213,103)(39,226,59,214,104)(40,227,60,215,105)(41,228,49,216,106)(42,217,50,205,107)(43,218,51,206,108)(44,219,52,207,97)(45,220,53,208,98)(46,221,54,209,99)(47,222,55,210,100)(48,223,56,211,101), (1,231)(2,236)(3,229)(4,234)(5,239)(6,232)(7,237)(8,230)(9,235)(10,240)(11,233)(12,238)(13,223)(14,228)(15,221)(16,226)(17,219)(18,224)(19,217)(20,222)(21,227)(22,220)(23,225)(24,218)(25,155)(26,148)(27,153)(28,146)(29,151)(30,156)(31,149)(32,154)(33,147)(34,152)(35,145)(36,150)(37,201)(38,194)(39,199)(40,204)(41,197)(42,202)(43,195)(44,200)(45,193)(46,198)(47,203)(48,196)(49,77)(50,82)(51,75)(52,80)(53,73)(54,78)(55,83)(56,76)(57,81)(58,74)(59,79)(60,84)(61,134)(62,139)(63,144)(64,137)(65,142)(66,135)(67,140)(68,133)(69,138)(70,143)(71,136)(72,141)(85,126)(86,131)(87,124)(88,129)(89,122)(90,127)(91,132)(92,125)(93,130)(94,123)(95,128)(96,121)(97,165)(98,158)(99,163)(100,168)(101,161)(102,166)(103,159)(104,164)(105,157)(106,162)(107,167)(108,160)(109,210)(110,215)(111,208)(112,213)(113,206)(114,211)(115,216)(116,209)(117,214)(118,207)(119,212)(120,205)(169,181)(170,186)(171,191)(172,184)(173,189)(174,182)(175,187)(176,192)(177,185)(178,190)(179,183)(180,188)>;
G:=Group( (1,221)(2,222)(3,223)(4,224)(5,225)(6,226)(7,227)(8,228)(9,217)(10,218)(11,219)(12,220)(13,229)(14,230)(15,231)(16,232)(17,233)(18,234)(19,235)(20,236)(21,237)(22,238)(23,239)(24,240)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,109)(33,110)(34,111)(35,112)(36,113)(37,182)(38,183)(39,184)(40,185)(41,186)(42,187)(43,188)(44,189)(45,190)(46,191)(47,192)(48,181)(49,130)(50,131)(51,132)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,127)(59,128)(60,129)(61,159)(62,160)(63,161)(64,162)(65,163)(66,164)(67,165)(68,166)(69,167)(70,168)(71,157)(72,158)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,85)(82,86)(83,87)(84,88)(97,140)(98,141)(99,142)(100,143)(101,144)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(145,213)(146,214)(147,215)(148,216)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,211)(156,212)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)(176,203)(177,204)(178,193)(179,194)(180,195), (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), (1,220)(2,219)(3,218)(4,217)(5,228)(6,227)(7,226)(8,225)(9,224)(10,223)(11,222)(12,221)(13,238)(14,237)(15,236)(16,235)(17,234)(18,233)(19,232)(20,231)(21,230)(22,229)(23,240)(24,239)(25,111)(26,110)(27,109)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,187)(38,186)(39,185)(40,184)(41,183)(42,182)(43,181)(44,192)(45,191)(46,190)(47,189)(48,188)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,132)(57,131)(58,130)(59,129)(60,128)(61,160)(62,159)(63,158)(64,157)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,96)(82,95)(83,94)(84,93)(97,143)(98,142)(99,141)(100,140)(101,139)(102,138)(103,137)(104,136)(105,135)(106,134)(107,133)(108,144)(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)(152,209)(153,208)(154,207)(155,206)(156,205)(169,193)(170,204)(171,203)(172,202)(173,201)(174,200)(175,199)(176,198)(177,197)(178,196)(179,195)(180,194), (1,123,153,142,191)(2,124,154,143,192)(3,125,155,144,181)(4,126,156,133,182)(5,127,145,134,183)(6,128,146,135,184)(7,129,147,136,185)(8,130,148,137,186)(9,131,149,138,187)(10,132,150,139,188)(11,121,151,140,189)(12,122,152,141,190)(13,196,161,114,76)(14,197,162,115,77)(15,198,163,116,78)(16,199,164,117,79)(17,200,165,118,80)(18,201,166,119,81)(19,202,167,120,82)(20,203,168,109,83)(21,204,157,110,84)(22,193,158,111,73)(23,194,159,112,74)(24,195,160,113,75)(25,92,229,169,63)(26,93,230,170,64)(27,94,231,171,65)(28,95,232,172,66)(29,96,233,173,67)(30,85,234,174,68)(31,86,235,175,69)(32,87,236,176,70)(33,88,237,177,71)(34,89,238,178,72)(35,90,239,179,61)(36,91,240,180,62)(37,224,57,212,102)(38,225,58,213,103)(39,226,59,214,104)(40,227,60,215,105)(41,228,49,216,106)(42,217,50,205,107)(43,218,51,206,108)(44,219,52,207,97)(45,220,53,208,98)(46,221,54,209,99)(47,222,55,210,100)(48,223,56,211,101), (1,231)(2,236)(3,229)(4,234)(5,239)(6,232)(7,237)(8,230)(9,235)(10,240)(11,233)(12,238)(13,223)(14,228)(15,221)(16,226)(17,219)(18,224)(19,217)(20,222)(21,227)(22,220)(23,225)(24,218)(25,155)(26,148)(27,153)(28,146)(29,151)(30,156)(31,149)(32,154)(33,147)(34,152)(35,145)(36,150)(37,201)(38,194)(39,199)(40,204)(41,197)(42,202)(43,195)(44,200)(45,193)(46,198)(47,203)(48,196)(49,77)(50,82)(51,75)(52,80)(53,73)(54,78)(55,83)(56,76)(57,81)(58,74)(59,79)(60,84)(61,134)(62,139)(63,144)(64,137)(65,142)(66,135)(67,140)(68,133)(69,138)(70,143)(71,136)(72,141)(85,126)(86,131)(87,124)(88,129)(89,122)(90,127)(91,132)(92,125)(93,130)(94,123)(95,128)(96,121)(97,165)(98,158)(99,163)(100,168)(101,161)(102,166)(103,159)(104,164)(105,157)(106,162)(107,167)(108,160)(109,210)(110,215)(111,208)(112,213)(113,206)(114,211)(115,216)(116,209)(117,214)(118,207)(119,212)(120,205)(169,181)(170,186)(171,191)(172,184)(173,189)(174,182)(175,187)(176,192)(177,185)(178,190)(179,183)(180,188) );
G=PermutationGroup([[(1,221),(2,222),(3,223),(4,224),(5,225),(6,226),(7,227),(8,228),(9,217),(10,218),(11,219),(12,220),(13,229),(14,230),(15,231),(16,232),(17,233),(18,234),(19,235),(20,236),(21,237),(22,238),(23,239),(24,240),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,109),(33,110),(34,111),(35,112),(36,113),(37,182),(38,183),(39,184),(40,185),(41,186),(42,187),(43,188),(44,189),(45,190),(46,191),(47,192),(48,181),(49,130),(50,131),(51,132),(52,121),(53,122),(54,123),(55,124),(56,125),(57,126),(58,127),(59,128),(60,129),(61,159),(62,160),(63,161),(64,162),(65,163),(66,164),(67,165),(68,166),(69,167),(70,168),(71,157),(72,158),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96),(81,85),(82,86),(83,87),(84,88),(97,140),(98,141),(99,142),(100,143),(101,144),(102,133),(103,134),(104,135),(105,136),(106,137),(107,138),(108,139),(145,213),(146,214),(147,215),(148,216),(149,205),(150,206),(151,207),(152,208),(153,209),(154,210),(155,211),(156,212),(169,196),(170,197),(171,198),(172,199),(173,200),(174,201),(175,202),(176,203),(177,204),(178,193),(179,194),(180,195)], [(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)], [(1,220),(2,219),(3,218),(4,217),(5,228),(6,227),(7,226),(8,225),(9,224),(10,223),(11,222),(12,221),(13,238),(14,237),(15,236),(16,235),(17,234),(18,233),(19,232),(20,231),(21,230),(22,229),(23,240),(24,239),(25,111),(26,110),(27,109),(28,120),(29,119),(30,118),(31,117),(32,116),(33,115),(34,114),(35,113),(36,112),(37,187),(38,186),(39,185),(40,184),(41,183),(42,182),(43,181),(44,192),(45,191),(46,190),(47,189),(48,188),(49,127),(50,126),(51,125),(52,124),(53,123),(54,122),(55,121),(56,132),(57,131),(58,130),(59,129),(60,128),(61,160),(62,159),(63,158),(64,157),(65,168),(66,167),(67,166),(68,165),(69,164),(70,163),(71,162),(72,161),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,96),(82,95),(83,94),(84,93),(97,143),(98,142),(99,141),(100,140),(101,139),(102,138),(103,137),(104,136),(105,135),(106,134),(107,133),(108,144),(145,216),(146,215),(147,214),(148,213),(149,212),(150,211),(151,210),(152,209),(153,208),(154,207),(155,206),(156,205),(169,193),(170,204),(171,203),(172,202),(173,201),(174,200),(175,199),(176,198),(177,197),(178,196),(179,195),(180,194)], [(1,123,153,142,191),(2,124,154,143,192),(3,125,155,144,181),(4,126,156,133,182),(5,127,145,134,183),(6,128,146,135,184),(7,129,147,136,185),(8,130,148,137,186),(9,131,149,138,187),(10,132,150,139,188),(11,121,151,140,189),(12,122,152,141,190),(13,196,161,114,76),(14,197,162,115,77),(15,198,163,116,78),(16,199,164,117,79),(17,200,165,118,80),(18,201,166,119,81),(19,202,167,120,82),(20,203,168,109,83),(21,204,157,110,84),(22,193,158,111,73),(23,194,159,112,74),(24,195,160,113,75),(25,92,229,169,63),(26,93,230,170,64),(27,94,231,171,65),(28,95,232,172,66),(29,96,233,173,67),(30,85,234,174,68),(31,86,235,175,69),(32,87,236,176,70),(33,88,237,177,71),(34,89,238,178,72),(35,90,239,179,61),(36,91,240,180,62),(37,224,57,212,102),(38,225,58,213,103),(39,226,59,214,104),(40,227,60,215,105),(41,228,49,216,106),(42,217,50,205,107),(43,218,51,206,108),(44,219,52,207,97),(45,220,53,208,98),(46,221,54,209,99),(47,222,55,210,100),(48,223,56,211,101)], [(1,231),(2,236),(3,229),(4,234),(5,239),(6,232),(7,237),(8,230),(9,235),(10,240),(11,233),(12,238),(13,223),(14,228),(15,221),(16,226),(17,219),(18,224),(19,217),(20,222),(21,227),(22,220),(23,225),(24,218),(25,155),(26,148),(27,153),(28,146),(29,151),(30,156),(31,149),(32,154),(33,147),(34,152),(35,145),(36,150),(37,201),(38,194),(39,199),(40,204),(41,197),(42,202),(43,195),(44,200),(45,193),(46,198),(47,203),(48,196),(49,77),(50,82),(51,75),(52,80),(53,73),(54,78),(55,83),(56,76),(57,81),(58,74),(59,79),(60,84),(61,134),(62,139),(63,144),(64,137),(65,142),(66,135),(67,140),(68,133),(69,138),(70,143),(71,136),(72,141),(85,126),(86,131),(87,124),(88,129),(89,122),(90,127),(91,132),(92,125),(93,130),(94,123),(95,128),(96,121),(97,165),(98,158),(99,163),(100,168),(101,161),(102,166),(103,159),(104,164),(105,157),(106,162),(107,167),(108,160),(109,210),(110,215),(111,208),(112,213),(113,206),(114,211),(115,216),(116,209),(117,214),(118,207),(119,212),(120,205),(169,181),(170,186),(171,191),(172,184),(173,189),(174,182),(175,187),(176,192),(177,185),(178,190),(179,183),(180,188)]])
66 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 | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 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 | 10 | ··· | 10 | 10 | ··· | 10 | 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 | 30 | 30 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | C2×S3×D5 | C2×S3×D5 | D12⋊D5 |
| kernel | C2×D12⋊D5 | D12⋊D5 | C2×S3×Dic5 | C2×C5⋊D12 | C6×Dic10 | C10×D12 | C2×C4×D15 | C2×Dic10 | C2×D12 | Dic10 | C2×Dic5 | C2×C20 | C30 | D12 | C2×C12 | C22×S3 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 4 | 8 | 2 | 4 | 2 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C2×D12⋊D5 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 52 |
| 0 | 0 | 0 | 0 | 15 | 45 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 17 |
| 0 | 0 | 0 | 0 | 46 | 16 |
| 18 | 60 | 0 | 0 | 0 | 0 |
| 19 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 53 | 0 | 0 | 0 | 0 |
| 14 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 23 |
| 0 | 0 | 0 | 0 | 43 | 7 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,16,15,0,0,0,0,52,45],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,45,46,0,0,0,0,17,16],[18,19,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,14,0,0,0,0,53,33,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,54,43,0,0,0,0,23,7] >;
C2×D12⋊D5 in GAP, Magma, Sage, TeX
C_2\times D_{12}\rtimes D_5 % in TeX
G:=Group("C2xD12:D5"); // GroupNames label
G:=SmallGroup(480,1079);
// by ID
G=gap.SmallGroup(480,1079);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,675,346,80,1356,18822]);
// 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,e*b*e=b^5,c*d=d*c,e*c*e=b^10*c,e*d*e=d^-1>;
// generators/relations