direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic6⋊D5, C30⋊6SD16, C60.65D4, C20.16D12, Dic6⋊17D10, C60.131C23, D60.50C22, C5⋊2C8⋊27D6, C6⋊1(Q8⋊D5), C15⋊9(C2×SD16), (C2×Dic6)⋊1D5, C30.92(C2×D4), (C2×C30).60D4, C10⋊2(C24⋊C2), (C10×Dic6)⋊4C2, (C2×D60).17C2, (C2×C10).41D12, (C2×C20).100D6, C10.52(C2×D12), C4.7(C5⋊D12), (C2×C12).291D10, C12.57(C5⋊D4), C20.97(C22×S3), (C2×C60).135C22, (C5×Dic6)⋊19C22, C12.154(C22×D5), C22.21(C5⋊D12), C3⋊1(C2×Q8⋊D5), C5⋊3(C2×C24⋊C2), (C6×C5⋊2C8)⋊8C2, C4.79(C2×S3×D5), (C2×C5⋊2C8)⋊8S3, C6.6(C2×C5⋊D4), (C2×C4).148(S3×D5), C2.10(C2×C5⋊D12), (C3×C5⋊2C8)⋊31C22, (C2×C6).33(C5⋊D4), SmallGroup(480,393)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic6⋊D5
G = < a,b,c,d,e | a2=b12=d5=e2=1, c2=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b3c, ede=d-1 >
Subgroups: 924 in 136 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, D10, C2×C10, C24, Dic6, Dic6, D12, C2×Dic3, C2×C12, C22×S3, D15, C30, C30, C2×SD16, C5⋊2C8, D20, C2×C20, C2×C20, C5×Q8, C22×D5, C24⋊C2, C2×C24, C2×Dic6, C2×D12, C5×Dic3, C60, D30, C2×C30, C2×C5⋊2C8, Q8⋊D5, C2×D20, Q8×C10, C2×C24⋊C2, C3×C5⋊2C8, C5×Dic6, C5×Dic6, C10×Dic3, D60, D60, C2×C60, C22×D15, C2×Q8⋊D5, Dic6⋊D5, C6×C5⋊2C8, C10×Dic6, C2×D60, C2×Dic6⋊D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, D12, C22×S3, C2×SD16, C5⋊D4, C22×D5, C24⋊C2, C2×D12, S3×D5, Q8⋊D5, C2×C5⋊D4, C2×C24⋊C2, C5⋊D12, C2×S3×D5, C2×Q8⋊D5, Dic6⋊D5, C2×C5⋊D12, C2×Dic6⋊D5
►On 240 points
Generators in S240
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 109)(11 110)(12 111)(13 174)(14 175)(15 176)(16 177)(17 178)(18 179)(19 180)(20 169)(21 170)(22 171)(23 172)(24 173)(25 226)(26 227)(27 228)(28 217)(29 218)(30 219)(31 220)(32 221)(33 222)(34 223)(35 224)(36 225)(37 72)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 193)(50 194)(51 195)(52 196)(53 197)(54 198)(55 199)(56 200)(57 201)(58 202)(59 203)(60 204)(73 129)(74 130)(75 131)(76 132)(77 121)(78 122)(79 123)(80 124)(81 125)(82 126)(83 127)(84 128)(85 233)(86 234)(87 235)(88 236)(89 237)(90 238)(91 239)(92 240)(93 229)(94 230)(95 231)(96 232)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)(145 208)(146 209)(147 210)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 205)(155 206)(156 207)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 190)(164 191)(165 192)(166 181)(167 182)(168 183)
(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 70 7 64)(2 69 8 63)(3 68 9 62)(4 67 10 61)(5 66 11 72)(6 65 12 71)(13 80 19 74)(14 79 20 73)(15 78 21 84)(16 77 22 83)(17 76 23 82)(18 75 24 81)(25 205 31 211)(26 216 32 210)(27 215 33 209)(28 214 34 208)(29 213 35 207)(30 212 36 206)(37 116 43 110)(38 115 44 109)(39 114 45 120)(40 113 46 119)(41 112 47 118)(42 111 48 117)(49 234 55 240)(50 233 56 239)(51 232 57 238)(52 231 58 237)(53 230 59 236)(54 229 60 235)(85 200 91 194)(86 199 92 193)(87 198 93 204)(88 197 94 203)(89 196 95 202)(90 195 96 201)(97 168 103 162)(98 167 104 161)(99 166 105 160)(100 165 106 159)(101 164 107 158)(102 163 108 157)(121 171 127 177)(122 170 128 176)(123 169 129 175)(124 180 130 174)(125 179 131 173)(126 178 132 172)(133 183 139 189)(134 182 140 188)(135 181 141 187)(136 192 142 186)(137 191 143 185)(138 190 144 184)(145 217 151 223)(146 228 152 222)(147 227 153 221)(148 226 154 220)(149 225 155 219)(150 224 156 218)
(1 102 80 207 86)(2 103 81 208 87)(3 104 82 209 88)(4 105 83 210 89)(5 106 84 211 90)(6 107 73 212 91)(7 108 74 213 92)(8 97 75 214 93)(9 98 76 215 94)(10 99 77 216 95)(11 100 78 205 96)(12 101 79 206 85)(13 35 193 64 157)(14 36 194 65 158)(15 25 195 66 159)(16 26 196 67 160)(17 27 197 68 161)(18 28 198 69 162)(19 29 199 70 163)(20 30 200 71 164)(21 31 201 72 165)(22 32 202 61 166)(23 33 203 62 167)(24 34 204 63 168)(37 192 170 220 57)(38 181 171 221 58)(39 182 172 222 59)(40 183 173 223 60)(41 184 174 224 49)(42 185 175 225 50)(43 186 176 226 51)(44 187 177 227 52)(45 188 178 228 53)(46 189 179 217 54)(47 190 180 218 55)(48 191 169 219 56)(109 135 121 153 231)(110 136 122 154 232)(111 137 123 155 233)(112 138 124 156 234)(113 139 125 145 235)(114 140 126 146 236)(115 141 127 147 237)(116 142 128 148 238)(117 143 129 149 239)(118 144 130 150 240)(119 133 131 151 229)(120 134 132 152 230)
(1 86)(2 85)(3 96)(4 95)(5 94)(6 93)(7 92)(8 91)(9 90)(10 89)(11 88)(12 87)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 164)(26 163)(27 162)(28 161)(29 160)(30 159)(31 158)(32 157)(33 168)(34 167)(35 166)(36 165)(37 50)(38 49)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(61 193)(62 204)(63 203)(64 202)(65 201)(66 200)(67 199)(68 198)(69 197)(70 196)(71 195)(72 194)(73 75)(76 84)(77 83)(78 82)(79 81)(97 212)(98 211)(99 210)(100 209)(101 208)(102 207)(103 206)(104 205)(105 216)(106 215)(107 214)(108 213)(109 237)(110 236)(111 235)(112 234)(113 233)(114 232)(115 231)(116 230)(117 229)(118 240)(119 239)(120 238)(121 127)(122 126)(123 125)(128 132)(129 131)(133 149)(134 148)(135 147)(136 146)(137 145)(138 156)(139 155)(140 154)(141 153)(142 152)(143 151)(144 150)(169 176)(170 175)(171 174)(172 173)(177 180)(178 179)(181 224)(182 223)(183 222)(184 221)(185 220)(186 219)(187 218)(188 217)(189 228)(190 227)(191 226)(192 225)
G:=sub<Sym(240)| (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,109)(11,110)(12,111)(13,174)(14,175)(15,176)(16,177)(17,178)(18,179)(19,180)(20,169)(21,170)(22,171)(23,172)(24,173)(25,226)(26,227)(27,228)(28,217)(29,218)(30,219)(31,220)(32,221)(33,222)(34,223)(35,224)(36,225)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,201)(58,202)(59,203)(60,204)(73,129)(74,130)(75,131)(76,132)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,127)(84,128)(85,233)(86,234)(87,235)(88,236)(89,237)(90,238)(91,239)(92,240)(93,229)(94,230)(95,231)(96,232)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,205)(155,206)(156,207)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,181)(167,182)(168,183), (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), (1,70,7,64)(2,69,8,63)(3,68,9,62)(4,67,10,61)(5,66,11,72)(6,65,12,71)(13,80,19,74)(14,79,20,73)(15,78,21,84)(16,77,22,83)(17,76,23,82)(18,75,24,81)(25,205,31,211)(26,216,32,210)(27,215,33,209)(28,214,34,208)(29,213,35,207)(30,212,36,206)(37,116,43,110)(38,115,44,109)(39,114,45,120)(40,113,46,119)(41,112,47,118)(42,111,48,117)(49,234,55,240)(50,233,56,239)(51,232,57,238)(52,231,58,237)(53,230,59,236)(54,229,60,235)(85,200,91,194)(86,199,92,193)(87,198,93,204)(88,197,94,203)(89,196,95,202)(90,195,96,201)(97,168,103,162)(98,167,104,161)(99,166,105,160)(100,165,106,159)(101,164,107,158)(102,163,108,157)(121,171,127,177)(122,170,128,176)(123,169,129,175)(124,180,130,174)(125,179,131,173)(126,178,132,172)(133,183,139,189)(134,182,140,188)(135,181,141,187)(136,192,142,186)(137,191,143,185)(138,190,144,184)(145,217,151,223)(146,228,152,222)(147,227,153,221)(148,226,154,220)(149,225,155,219)(150,224,156,218), (1,102,80,207,86)(2,103,81,208,87)(3,104,82,209,88)(4,105,83,210,89)(5,106,84,211,90)(6,107,73,212,91)(7,108,74,213,92)(8,97,75,214,93)(9,98,76,215,94)(10,99,77,216,95)(11,100,78,205,96)(12,101,79,206,85)(13,35,193,64,157)(14,36,194,65,158)(15,25,195,66,159)(16,26,196,67,160)(17,27,197,68,161)(18,28,198,69,162)(19,29,199,70,163)(20,30,200,71,164)(21,31,201,72,165)(22,32,202,61,166)(23,33,203,62,167)(24,34,204,63,168)(37,192,170,220,57)(38,181,171,221,58)(39,182,172,222,59)(40,183,173,223,60)(41,184,174,224,49)(42,185,175,225,50)(43,186,176,226,51)(44,187,177,227,52)(45,188,178,228,53)(46,189,179,217,54)(47,190,180,218,55)(48,191,169,219,56)(109,135,121,153,231)(110,136,122,154,232)(111,137,123,155,233)(112,138,124,156,234)(113,139,125,145,235)(114,140,126,146,236)(115,141,127,147,237)(116,142,128,148,238)(117,143,129,149,239)(118,144,130,150,240)(119,133,131,151,229)(120,134,132,152,230), (1,86)(2,85)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,164)(26,163)(27,162)(28,161)(29,160)(30,159)(31,158)(32,157)(33,168)(34,167)(35,166)(36,165)(37,50)(38,49)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(61,193)(62,204)(63,203)(64,202)(65,201)(66,200)(67,199)(68,198)(69,197)(70,196)(71,195)(72,194)(73,75)(76,84)(77,83)(78,82)(79,81)(97,212)(98,211)(99,210)(100,209)(101,208)(102,207)(103,206)(104,205)(105,216)(106,215)(107,214)(108,213)(109,237)(110,236)(111,235)(112,234)(113,233)(114,232)(115,231)(116,230)(117,229)(118,240)(119,239)(120,238)(121,127)(122,126)(123,125)(128,132)(129,131)(133,149)(134,148)(135,147)(136,146)(137,145)(138,156)(139,155)(140,154)(141,153)(142,152)(143,151)(144,150)(169,176)(170,175)(171,174)(172,173)(177,180)(178,179)(181,224)(182,223)(183,222)(184,221)(185,220)(186,219)(187,218)(188,217)(189,228)(190,227)(191,226)(192,225)>;
G:=Group( (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,109)(11,110)(12,111)(13,174)(14,175)(15,176)(16,177)(17,178)(18,179)(19,180)(20,169)(21,170)(22,171)(23,172)(24,173)(25,226)(26,227)(27,228)(28,217)(29,218)(30,219)(31,220)(32,221)(33,222)(34,223)(35,224)(36,225)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,201)(58,202)(59,203)(60,204)(73,129)(74,130)(75,131)(76,132)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,127)(84,128)(85,233)(86,234)(87,235)(88,236)(89,237)(90,238)(91,239)(92,240)(93,229)(94,230)(95,231)(96,232)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,205)(155,206)(156,207)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,181)(167,182)(168,183), (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), (1,70,7,64)(2,69,8,63)(3,68,9,62)(4,67,10,61)(5,66,11,72)(6,65,12,71)(13,80,19,74)(14,79,20,73)(15,78,21,84)(16,77,22,83)(17,76,23,82)(18,75,24,81)(25,205,31,211)(26,216,32,210)(27,215,33,209)(28,214,34,208)(29,213,35,207)(30,212,36,206)(37,116,43,110)(38,115,44,109)(39,114,45,120)(40,113,46,119)(41,112,47,118)(42,111,48,117)(49,234,55,240)(50,233,56,239)(51,232,57,238)(52,231,58,237)(53,230,59,236)(54,229,60,235)(85,200,91,194)(86,199,92,193)(87,198,93,204)(88,197,94,203)(89,196,95,202)(90,195,96,201)(97,168,103,162)(98,167,104,161)(99,166,105,160)(100,165,106,159)(101,164,107,158)(102,163,108,157)(121,171,127,177)(122,170,128,176)(123,169,129,175)(124,180,130,174)(125,179,131,173)(126,178,132,172)(133,183,139,189)(134,182,140,188)(135,181,141,187)(136,192,142,186)(137,191,143,185)(138,190,144,184)(145,217,151,223)(146,228,152,222)(147,227,153,221)(148,226,154,220)(149,225,155,219)(150,224,156,218), (1,102,80,207,86)(2,103,81,208,87)(3,104,82,209,88)(4,105,83,210,89)(5,106,84,211,90)(6,107,73,212,91)(7,108,74,213,92)(8,97,75,214,93)(9,98,76,215,94)(10,99,77,216,95)(11,100,78,205,96)(12,101,79,206,85)(13,35,193,64,157)(14,36,194,65,158)(15,25,195,66,159)(16,26,196,67,160)(17,27,197,68,161)(18,28,198,69,162)(19,29,199,70,163)(20,30,200,71,164)(21,31,201,72,165)(22,32,202,61,166)(23,33,203,62,167)(24,34,204,63,168)(37,192,170,220,57)(38,181,171,221,58)(39,182,172,222,59)(40,183,173,223,60)(41,184,174,224,49)(42,185,175,225,50)(43,186,176,226,51)(44,187,177,227,52)(45,188,178,228,53)(46,189,179,217,54)(47,190,180,218,55)(48,191,169,219,56)(109,135,121,153,231)(110,136,122,154,232)(111,137,123,155,233)(112,138,124,156,234)(113,139,125,145,235)(114,140,126,146,236)(115,141,127,147,237)(116,142,128,148,238)(117,143,129,149,239)(118,144,130,150,240)(119,133,131,151,229)(120,134,132,152,230), (1,86)(2,85)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,164)(26,163)(27,162)(28,161)(29,160)(30,159)(31,158)(32,157)(33,168)(34,167)(35,166)(36,165)(37,50)(38,49)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(61,193)(62,204)(63,203)(64,202)(65,201)(66,200)(67,199)(68,198)(69,197)(70,196)(71,195)(72,194)(73,75)(76,84)(77,83)(78,82)(79,81)(97,212)(98,211)(99,210)(100,209)(101,208)(102,207)(103,206)(104,205)(105,216)(106,215)(107,214)(108,213)(109,237)(110,236)(111,235)(112,234)(113,233)(114,232)(115,231)(116,230)(117,229)(118,240)(119,239)(120,238)(121,127)(122,126)(123,125)(128,132)(129,131)(133,149)(134,148)(135,147)(136,146)(137,145)(138,156)(139,155)(140,154)(141,153)(142,152)(143,151)(144,150)(169,176)(170,175)(171,174)(172,173)(177,180)(178,179)(181,224)(182,223)(183,222)(184,221)(185,220)(186,219)(187,218)(188,217)(189,228)(190,227)(191,226)(192,225) );
G=PermutationGroup([[(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,118),(8,119),(9,120),(10,109),(11,110),(12,111),(13,174),(14,175),(15,176),(16,177),(17,178),(18,179),(19,180),(20,169),(21,170),(22,171),(23,172),(24,173),(25,226),(26,227),(27,228),(28,217),(29,218),(30,219),(31,220),(32,221),(33,222),(34,223),(35,224),(36,225),(37,72),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,193),(50,194),(51,195),(52,196),(53,197),(54,198),(55,199),(56,200),(57,201),(58,202),(59,203),(60,204),(73,129),(74,130),(75,131),(76,132),(77,121),(78,122),(79,123),(80,124),(81,125),(82,126),(83,127),(84,128),(85,233),(86,234),(87,235),(88,236),(89,237),(90,238),(91,239),(92,240),(93,229),(94,230),(95,231),(96,232),(97,133),(98,134),(99,135),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144),(145,208),(146,209),(147,210),(148,211),(149,212),(150,213),(151,214),(152,215),(153,216),(154,205),(155,206),(156,207),(157,184),(158,185),(159,186),(160,187),(161,188),(162,189),(163,190),(164,191),(165,192),(166,181),(167,182),(168,183)], [(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)], [(1,70,7,64),(2,69,8,63),(3,68,9,62),(4,67,10,61),(5,66,11,72),(6,65,12,71),(13,80,19,74),(14,79,20,73),(15,78,21,84),(16,77,22,83),(17,76,23,82),(18,75,24,81),(25,205,31,211),(26,216,32,210),(27,215,33,209),(28,214,34,208),(29,213,35,207),(30,212,36,206),(37,116,43,110),(38,115,44,109),(39,114,45,120),(40,113,46,119),(41,112,47,118),(42,111,48,117),(49,234,55,240),(50,233,56,239),(51,232,57,238),(52,231,58,237),(53,230,59,236),(54,229,60,235),(85,200,91,194),(86,199,92,193),(87,198,93,204),(88,197,94,203),(89,196,95,202),(90,195,96,201),(97,168,103,162),(98,167,104,161),(99,166,105,160),(100,165,106,159),(101,164,107,158),(102,163,108,157),(121,171,127,177),(122,170,128,176),(123,169,129,175),(124,180,130,174),(125,179,131,173),(126,178,132,172),(133,183,139,189),(134,182,140,188),(135,181,141,187),(136,192,142,186),(137,191,143,185),(138,190,144,184),(145,217,151,223),(146,228,152,222),(147,227,153,221),(148,226,154,220),(149,225,155,219),(150,224,156,218)], [(1,102,80,207,86),(2,103,81,208,87),(3,104,82,209,88),(4,105,83,210,89),(5,106,84,211,90),(6,107,73,212,91),(7,108,74,213,92),(8,97,75,214,93),(9,98,76,215,94),(10,99,77,216,95),(11,100,78,205,96),(12,101,79,206,85),(13,35,193,64,157),(14,36,194,65,158),(15,25,195,66,159),(16,26,196,67,160),(17,27,197,68,161),(18,28,198,69,162),(19,29,199,70,163),(20,30,200,71,164),(21,31,201,72,165),(22,32,202,61,166),(23,33,203,62,167),(24,34,204,63,168),(37,192,170,220,57),(38,181,171,221,58),(39,182,172,222,59),(40,183,173,223,60),(41,184,174,224,49),(42,185,175,225,50),(43,186,176,226,51),(44,187,177,227,52),(45,188,178,228,53),(46,189,179,217,54),(47,190,180,218,55),(48,191,169,219,56),(109,135,121,153,231),(110,136,122,154,232),(111,137,123,155,233),(112,138,124,156,234),(113,139,125,145,235),(114,140,126,146,236),(115,141,127,147,237),(116,142,128,148,238),(117,143,129,149,239),(118,144,130,150,240),(119,133,131,151,229),(120,134,132,152,230)], [(1,86),(2,85),(3,96),(4,95),(5,94),(6,93),(7,92),(8,91),(9,90),(10,89),(11,88),(12,87),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(25,164),(26,163),(27,162),(28,161),(29,160),(30,159),(31,158),(32,157),(33,168),(34,167),(35,166),(36,165),(37,50),(38,49),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(61,193),(62,204),(63,203),(64,202),(65,201),(66,200),(67,199),(68,198),(69,197),(70,196),(71,195),(72,194),(73,75),(76,84),(77,83),(78,82),(79,81),(97,212),(98,211),(99,210),(100,209),(101,208),(102,207),(103,206),(104,205),(105,216),(106,215),(107,214),(108,213),(109,237),(110,236),(111,235),(112,234),(113,233),(114,232),(115,231),(116,230),(117,229),(118,240),(119,239),(120,238),(121,127),(122,126),(123,125),(128,132),(129,131),(133,149),(134,148),(135,147),(136,146),(137,145),(138,156),(139,155),(140,154),(141,153),(142,152),(143,151),(144,150),(169,176),(170,175),(171,174),(172,173),(177,180),(178,179),(181,224),(182,223),(183,222),(184,221),(185,220),(186,219),(187,218),(188,217),(189,228),(190,227),(191,226),(192,225)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | SD16 | D10 | D10 | D12 | D12 | C5⋊D4 | C5⋊D4 | C24⋊C2 | S3×D5 | Q8⋊D5 | C5⋊D12 | C2×S3×D5 | C5⋊D12 | Dic6⋊D5 |
| kernel | C2×Dic6⋊D5 | Dic6⋊D5 | C6×C5⋊2C8 | C10×Dic6 | C2×D60 | C2×C5⋊2C8 | C60 | C2×C30 | C2×Dic6 | C5⋊2C8 | C2×C20 | C30 | Dic6 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C2×Dic6⋊D5 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 239 | 49 | 0 | 0 | 0 | 0 |
| 177 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 103 | 5 |
| 0 | 0 | 0 | 0 | 47 | 138 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 177 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 193 | 146 |
| 0 | 0 | 0 | 0 | 75 | 48 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 64 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 190 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 | 240 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[239,177,0,0,0,0,49,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,47,0,0,0,0,5,138],[240,177,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,193,75,0,0,0,0,146,48],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,1,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,64,0,0,0,0,0,240,0,0,0,0,0,0,51,190,0,0,0,0,240,190,0,0,0,0,0,0,1,7,0,0,0,0,0,240] >;
C2×Dic6⋊D5 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_6\rtimes D_5 % in TeX
G:=Group("C2xDic6:D5"); // GroupNames label
G:=SmallGroup(480,393);
// by ID
G=gap.SmallGroup(480,393);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,120,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^12=d^5=e^2=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations