metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×Dic6)⋊6D5, C30.150(C2×D4), (C2×C20).231D6, (C2×C12).27D10, C10.137(S3×D4), D10⋊C4⋊18S3, D30⋊3C4⋊21C2, D30⋊4C4⋊19C2, (C10×Dic6)⋊16C2, C30.89(C4○D4), C3⋊2(C20.23D4), C15⋊10(C4.4D4), (C5×Dic3).10D4, (C22×D5).16D6, (Dic3×Dic5)⋊23C2, C10.16(C4○D12), D10⋊Dic3⋊16C2, (C2×C60).324C22, (C2×C30).145C23, C6.18(Q8⋊2D5), (C2×Dic5).119D6, Dic3.2(C5⋊D4), C5⋊6(C23.11D6), C10.15(D4⋊2S3), C2.18(C12.28D10), C2.18(D20⋊S3), (C2×Dic3).113D10, (C6×Dic5).87C22, (C10×Dic3).88C22, (C22×D15).50C22, (C2×Dic15).112C22, (C2×C4).60(S3×D5), C2.18(S3×C5⋊D4), C6.39(C2×C5⋊D4), (C2×C3⋊D20).9C2, (D5×C2×C6).30C22, C22.197(C2×S3×D5), (C3×D10⋊C4)⋊23C2, (C2×C6).157(C22×D5), (C2×C10).157(C22×S3), SmallGroup(480,531)
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=b-1, bd=db, ebe=ab7, cd=dc, ece=b6c, ede=d-1 >
Subgroups: 876 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C4.4D4, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, D30, C2×C30, C4×Dic5, D10⋊C4, D10⋊C4, C2×D20, Q8×C10, C23.11D6, C3⋊D20, C6×Dic5, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C20.23D4, Dic3×Dic5, D10⋊Dic3, D30⋊4C4, C3×D10⋊C4, D30⋊3C4, C2×C3⋊D20, C10×Dic6, (C2×Dic6)⋊D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4.4D4, C5⋊D4, C22×D5, C4○D12, S3×D4, D4⋊2S3, S3×D5, Q8⋊2D5, C2×C5⋊D4, C23.11D6, C2×S3×D5, C20.23D4, D20⋊S3, C12.28D10, S3×C5⋊D4, (C2×Dic6)⋊D5
►On 240 points
Generators in S240
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 97)(11 98)(12 99)(13 228)(14 217)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(22 225)(23 226)(24 227)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 133)(36 134)(37 229)(38 230)(39 231)(40 232)(41 233)(42 234)(43 235)(44 236)(45 237)(46 238)(47 239)(48 240)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 189)(62 190)(63 191)(64 192)(65 181)(66 182)(67 183)(68 184)(69 185)(70 186)(71 187)(72 188)(85 193)(86 194)(87 195)(88 196)(89 197)(90 198)(91 199)(92 200)(93 201)(94 202)(95 203)(96 204)(109 161)(110 162)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 157)(118 158)(119 159)(120 160)(121 214)(122 215)(123 216)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)(145 177)(146 178)(147 179)(148 180)(149 169)(150 170)(151 171)(152 172)(153 173)(154 174)(155 175)(156 176)
(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 89 7 95)(2 88 8 94)(3 87 9 93)(4 86 10 92)(5 85 11 91)(6 96 12 90)(13 121 19 127)(14 132 20 126)(15 131 21 125)(16 130 22 124)(17 129 23 123)(18 128 24 122)(25 84 31 78)(26 83 32 77)(27 82 33 76)(28 81 34 75)(29 80 35 74)(30 79 36 73)(37 117 43 111)(38 116 44 110)(39 115 45 109)(40 114 46 120)(41 113 47 119)(42 112 48 118)(49 134 55 140)(50 133 56 139)(51 144 57 138)(52 143 58 137)(53 142 59 136)(54 141 60 135)(61 145 67 151)(62 156 68 150)(63 155 69 149)(64 154 70 148)(65 153 71 147)(66 152 72 146)(97 200 103 194)(98 199 104 193)(99 198 105 204)(100 197 106 203)(101 196 107 202)(102 195 108 201)(157 235 163 229)(158 234 164 240)(159 233 165 239)(160 232 166 238)(161 231 167 237)(162 230 168 236)(169 191 175 185)(170 190 176 184)(171 189 177 183)(172 188 178 182)(173 187 179 181)(174 186 180 192)(205 219 211 225)(206 218 212 224)(207 217 213 223)(208 228 214 222)(209 227 215 221)(210 226 216 220)
(1 155 233 33 223)(2 156 234 34 224)(3 145 235 35 225)(4 146 236 36 226)(5 147 237 25 227)(6 148 238 26 228)(7 149 239 27 217)(8 150 240 28 218)(9 151 229 29 219)(10 152 230 30 220)(11 153 231 31 221)(12 154 232 32 222)(13 105 180 46 136)(14 106 169 47 137)(15 107 170 48 138)(16 108 171 37 139)(17 97 172 38 140)(18 98 173 39 141)(19 99 174 40 142)(20 100 175 41 143)(21 101 176 42 144)(22 102 177 43 133)(23 103 178 44 134)(24 104 179 45 135)(49 129 200 188 116)(50 130 201 189 117)(51 131 202 190 118)(52 132 203 191 119)(53 121 204 192 120)(54 122 193 181 109)(55 123 194 182 110)(56 124 195 183 111)(57 125 196 184 112)(58 126 197 185 113)(59 127 198 186 114)(60 128 199 187 115)(61 157 80 211 93)(62 158 81 212 94)(63 159 82 213 95)(64 160 83 214 96)(65 161 84 215 85)(66 162 73 216 86)(67 163 74 205 87)(68 164 75 206 88)(69 165 76 207 89)(70 166 77 208 90)(71 167 78 209 91)(72 168 79 210 92)
(1 213)(2 127)(3 215)(4 129)(5 205)(6 131)(7 207)(8 121)(9 209)(10 123)(11 211)(12 125)(13 94)(14 197)(15 96)(16 199)(17 86)(18 201)(19 88)(20 203)(21 90)(22 193)(23 92)(24 195)(25 67)(26 190)(27 69)(28 192)(29 71)(30 182)(31 61)(32 184)(33 63)(34 186)(35 65)(36 188)(37 115)(38 162)(39 117)(40 164)(41 119)(42 166)(43 109)(44 168)(45 111)(46 158)(47 113)(48 160)(49 146)(50 173)(51 148)(52 175)(53 150)(54 177)(55 152)(56 179)(57 154)(58 169)(59 156)(60 171)(62 136)(64 138)(66 140)(68 142)(70 144)(72 134)(73 172)(74 147)(75 174)(76 149)(77 176)(78 151)(79 178)(80 153)(81 180)(82 155)(83 170)(84 145)(85 225)(87 227)(89 217)(91 219)(93 221)(95 223)(97 216)(98 130)(99 206)(100 132)(101 208)(102 122)(103 210)(104 124)(105 212)(106 126)(107 214)(108 128)(110 230)(112 232)(114 234)(116 236)(118 238)(120 240)(133 181)(135 183)(137 185)(139 187)(141 189)(143 191)(157 231)(159 233)(161 235)(163 237)(165 239)(167 229)(194 220)(196 222)(198 224)(200 226)(202 228)(204 218)
G:=sub<Sym(240)| (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,97)(11,98)(12,99)(13,228)(14,217)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,225)(23,226)(24,227)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,229)(38,230)(39,231)(40,232)(41,233)(42,234)(43,235)(44,236)(45,237)(46,238)(47,239)(48,240)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,189)(62,190)(63,191)(64,192)(65,181)(66,182)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(85,193)(86,194)(87,195)(88,196)(89,197)(90,198)(91,199)(92,200)(93,201)(94,202)(95,203)(96,204)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,157)(118,158)(119,159)(120,160)(121,214)(122,215)(123,216)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(145,177)(146,178)(147,179)(148,180)(149,169)(150,170)(151,171)(152,172)(153,173)(154,174)(155,175)(156,176), (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), (1,89,7,95)(2,88,8,94)(3,87,9,93)(4,86,10,92)(5,85,11,91)(6,96,12,90)(13,121,19,127)(14,132,20,126)(15,131,21,125)(16,130,22,124)(17,129,23,123)(18,128,24,122)(25,84,31,78)(26,83,32,77)(27,82,33,76)(28,81,34,75)(29,80,35,74)(30,79,36,73)(37,117,43,111)(38,116,44,110)(39,115,45,109)(40,114,46,120)(41,113,47,119)(42,112,48,118)(49,134,55,140)(50,133,56,139)(51,144,57,138)(52,143,58,137)(53,142,59,136)(54,141,60,135)(61,145,67,151)(62,156,68,150)(63,155,69,149)(64,154,70,148)(65,153,71,147)(66,152,72,146)(97,200,103,194)(98,199,104,193)(99,198,105,204)(100,197,106,203)(101,196,107,202)(102,195,108,201)(157,235,163,229)(158,234,164,240)(159,233,165,239)(160,232,166,238)(161,231,167,237)(162,230,168,236)(169,191,175,185)(170,190,176,184)(171,189,177,183)(172,188,178,182)(173,187,179,181)(174,186,180,192)(205,219,211,225)(206,218,212,224)(207,217,213,223)(208,228,214,222)(209,227,215,221)(210,226,216,220), (1,155,233,33,223)(2,156,234,34,224)(3,145,235,35,225)(4,146,236,36,226)(5,147,237,25,227)(6,148,238,26,228)(7,149,239,27,217)(8,150,240,28,218)(9,151,229,29,219)(10,152,230,30,220)(11,153,231,31,221)(12,154,232,32,222)(13,105,180,46,136)(14,106,169,47,137)(15,107,170,48,138)(16,108,171,37,139)(17,97,172,38,140)(18,98,173,39,141)(19,99,174,40,142)(20,100,175,41,143)(21,101,176,42,144)(22,102,177,43,133)(23,103,178,44,134)(24,104,179,45,135)(49,129,200,188,116)(50,130,201,189,117)(51,131,202,190,118)(52,132,203,191,119)(53,121,204,192,120)(54,122,193,181,109)(55,123,194,182,110)(56,124,195,183,111)(57,125,196,184,112)(58,126,197,185,113)(59,127,198,186,114)(60,128,199,187,115)(61,157,80,211,93)(62,158,81,212,94)(63,159,82,213,95)(64,160,83,214,96)(65,161,84,215,85)(66,162,73,216,86)(67,163,74,205,87)(68,164,75,206,88)(69,165,76,207,89)(70,166,77,208,90)(71,167,78,209,91)(72,168,79,210,92), (1,213)(2,127)(3,215)(4,129)(5,205)(6,131)(7,207)(8,121)(9,209)(10,123)(11,211)(12,125)(13,94)(14,197)(15,96)(16,199)(17,86)(18,201)(19,88)(20,203)(21,90)(22,193)(23,92)(24,195)(25,67)(26,190)(27,69)(28,192)(29,71)(30,182)(31,61)(32,184)(33,63)(34,186)(35,65)(36,188)(37,115)(38,162)(39,117)(40,164)(41,119)(42,166)(43,109)(44,168)(45,111)(46,158)(47,113)(48,160)(49,146)(50,173)(51,148)(52,175)(53,150)(54,177)(55,152)(56,179)(57,154)(58,169)(59,156)(60,171)(62,136)(64,138)(66,140)(68,142)(70,144)(72,134)(73,172)(74,147)(75,174)(76,149)(77,176)(78,151)(79,178)(80,153)(81,180)(82,155)(83,170)(84,145)(85,225)(87,227)(89,217)(91,219)(93,221)(95,223)(97,216)(98,130)(99,206)(100,132)(101,208)(102,122)(103,210)(104,124)(105,212)(106,126)(107,214)(108,128)(110,230)(112,232)(114,234)(116,236)(118,238)(120,240)(133,181)(135,183)(137,185)(139,187)(141,189)(143,191)(157,231)(159,233)(161,235)(163,237)(165,239)(167,229)(194,220)(196,222)(198,224)(200,226)(202,228)(204,218)>;
G:=Group( (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,97)(11,98)(12,99)(13,228)(14,217)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,225)(23,226)(24,227)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,229)(38,230)(39,231)(40,232)(41,233)(42,234)(43,235)(44,236)(45,237)(46,238)(47,239)(48,240)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,189)(62,190)(63,191)(64,192)(65,181)(66,182)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(85,193)(86,194)(87,195)(88,196)(89,197)(90,198)(91,199)(92,200)(93,201)(94,202)(95,203)(96,204)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,157)(118,158)(119,159)(120,160)(121,214)(122,215)(123,216)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(145,177)(146,178)(147,179)(148,180)(149,169)(150,170)(151,171)(152,172)(153,173)(154,174)(155,175)(156,176), (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), (1,89,7,95)(2,88,8,94)(3,87,9,93)(4,86,10,92)(5,85,11,91)(6,96,12,90)(13,121,19,127)(14,132,20,126)(15,131,21,125)(16,130,22,124)(17,129,23,123)(18,128,24,122)(25,84,31,78)(26,83,32,77)(27,82,33,76)(28,81,34,75)(29,80,35,74)(30,79,36,73)(37,117,43,111)(38,116,44,110)(39,115,45,109)(40,114,46,120)(41,113,47,119)(42,112,48,118)(49,134,55,140)(50,133,56,139)(51,144,57,138)(52,143,58,137)(53,142,59,136)(54,141,60,135)(61,145,67,151)(62,156,68,150)(63,155,69,149)(64,154,70,148)(65,153,71,147)(66,152,72,146)(97,200,103,194)(98,199,104,193)(99,198,105,204)(100,197,106,203)(101,196,107,202)(102,195,108,201)(157,235,163,229)(158,234,164,240)(159,233,165,239)(160,232,166,238)(161,231,167,237)(162,230,168,236)(169,191,175,185)(170,190,176,184)(171,189,177,183)(172,188,178,182)(173,187,179,181)(174,186,180,192)(205,219,211,225)(206,218,212,224)(207,217,213,223)(208,228,214,222)(209,227,215,221)(210,226,216,220), (1,155,233,33,223)(2,156,234,34,224)(3,145,235,35,225)(4,146,236,36,226)(5,147,237,25,227)(6,148,238,26,228)(7,149,239,27,217)(8,150,240,28,218)(9,151,229,29,219)(10,152,230,30,220)(11,153,231,31,221)(12,154,232,32,222)(13,105,180,46,136)(14,106,169,47,137)(15,107,170,48,138)(16,108,171,37,139)(17,97,172,38,140)(18,98,173,39,141)(19,99,174,40,142)(20,100,175,41,143)(21,101,176,42,144)(22,102,177,43,133)(23,103,178,44,134)(24,104,179,45,135)(49,129,200,188,116)(50,130,201,189,117)(51,131,202,190,118)(52,132,203,191,119)(53,121,204,192,120)(54,122,193,181,109)(55,123,194,182,110)(56,124,195,183,111)(57,125,196,184,112)(58,126,197,185,113)(59,127,198,186,114)(60,128,199,187,115)(61,157,80,211,93)(62,158,81,212,94)(63,159,82,213,95)(64,160,83,214,96)(65,161,84,215,85)(66,162,73,216,86)(67,163,74,205,87)(68,164,75,206,88)(69,165,76,207,89)(70,166,77,208,90)(71,167,78,209,91)(72,168,79,210,92), (1,213)(2,127)(3,215)(4,129)(5,205)(6,131)(7,207)(8,121)(9,209)(10,123)(11,211)(12,125)(13,94)(14,197)(15,96)(16,199)(17,86)(18,201)(19,88)(20,203)(21,90)(22,193)(23,92)(24,195)(25,67)(26,190)(27,69)(28,192)(29,71)(30,182)(31,61)(32,184)(33,63)(34,186)(35,65)(36,188)(37,115)(38,162)(39,117)(40,164)(41,119)(42,166)(43,109)(44,168)(45,111)(46,158)(47,113)(48,160)(49,146)(50,173)(51,148)(52,175)(53,150)(54,177)(55,152)(56,179)(57,154)(58,169)(59,156)(60,171)(62,136)(64,138)(66,140)(68,142)(70,144)(72,134)(73,172)(74,147)(75,174)(76,149)(77,176)(78,151)(79,178)(80,153)(81,180)(82,155)(83,170)(84,145)(85,225)(87,227)(89,217)(91,219)(93,221)(95,223)(97,216)(98,130)(99,206)(100,132)(101,208)(102,122)(103,210)(104,124)(105,212)(106,126)(107,214)(108,128)(110,230)(112,232)(114,234)(116,236)(118,238)(120,240)(133,181)(135,183)(137,185)(139,187)(141,189)(143,191)(157,231)(159,233)(161,235)(163,237)(165,239)(167,229)(194,220)(196,222)(198,224)(200,226)(202,228)(204,218) );
G=PermutationGroup([[(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,97),(11,98),(12,99),(13,228),(14,217),(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(22,225),(23,226),(24,227),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,133),(36,134),(37,229),(38,230),(39,231),(40,232),(41,233),(42,234),(43,235),(44,236),(45,237),(46,238),(47,239),(48,240),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,189),(62,190),(63,191),(64,192),(65,181),(66,182),(67,183),(68,184),(69,185),(70,186),(71,187),(72,188),(85,193),(86,194),(87,195),(88,196),(89,197),(90,198),(91,199),(92,200),(93,201),(94,202),(95,203),(96,204),(109,161),(110,162),(111,163),(112,164),(113,165),(114,166),(115,167),(116,168),(117,157),(118,158),(119,159),(120,160),(121,214),(122,215),(123,216),(124,205),(125,206),(126,207),(127,208),(128,209),(129,210),(130,211),(131,212),(132,213),(145,177),(146,178),(147,179),(148,180),(149,169),(150,170),(151,171),(152,172),(153,173),(154,174),(155,175),(156,176)], [(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)], [(1,89,7,95),(2,88,8,94),(3,87,9,93),(4,86,10,92),(5,85,11,91),(6,96,12,90),(13,121,19,127),(14,132,20,126),(15,131,21,125),(16,130,22,124),(17,129,23,123),(18,128,24,122),(25,84,31,78),(26,83,32,77),(27,82,33,76),(28,81,34,75),(29,80,35,74),(30,79,36,73),(37,117,43,111),(38,116,44,110),(39,115,45,109),(40,114,46,120),(41,113,47,119),(42,112,48,118),(49,134,55,140),(50,133,56,139),(51,144,57,138),(52,143,58,137),(53,142,59,136),(54,141,60,135),(61,145,67,151),(62,156,68,150),(63,155,69,149),(64,154,70,148),(65,153,71,147),(66,152,72,146),(97,200,103,194),(98,199,104,193),(99,198,105,204),(100,197,106,203),(101,196,107,202),(102,195,108,201),(157,235,163,229),(158,234,164,240),(159,233,165,239),(160,232,166,238),(161,231,167,237),(162,230,168,236),(169,191,175,185),(170,190,176,184),(171,189,177,183),(172,188,178,182),(173,187,179,181),(174,186,180,192),(205,219,211,225),(206,218,212,224),(207,217,213,223),(208,228,214,222),(209,227,215,221),(210,226,216,220)], [(1,155,233,33,223),(2,156,234,34,224),(3,145,235,35,225),(4,146,236,36,226),(5,147,237,25,227),(6,148,238,26,228),(7,149,239,27,217),(8,150,240,28,218),(9,151,229,29,219),(10,152,230,30,220),(11,153,231,31,221),(12,154,232,32,222),(13,105,180,46,136),(14,106,169,47,137),(15,107,170,48,138),(16,108,171,37,139),(17,97,172,38,140),(18,98,173,39,141),(19,99,174,40,142),(20,100,175,41,143),(21,101,176,42,144),(22,102,177,43,133),(23,103,178,44,134),(24,104,179,45,135),(49,129,200,188,116),(50,130,201,189,117),(51,131,202,190,118),(52,132,203,191,119),(53,121,204,192,120),(54,122,193,181,109),(55,123,194,182,110),(56,124,195,183,111),(57,125,196,184,112),(58,126,197,185,113),(59,127,198,186,114),(60,128,199,187,115),(61,157,80,211,93),(62,158,81,212,94),(63,159,82,213,95),(64,160,83,214,96),(65,161,84,215,85),(66,162,73,216,86),(67,163,74,205,87),(68,164,75,206,88),(69,165,76,207,89),(70,166,77,208,90),(71,167,78,209,91),(72,168,79,210,92)], [(1,213),(2,127),(3,215),(4,129),(5,205),(6,131),(7,207),(8,121),(9,209),(10,123),(11,211),(12,125),(13,94),(14,197),(15,96),(16,199),(17,86),(18,201),(19,88),(20,203),(21,90),(22,193),(23,92),(24,195),(25,67),(26,190),(27,69),(28,192),(29,71),(30,182),(31,61),(32,184),(33,63),(34,186),(35,65),(36,188),(37,115),(38,162),(39,117),(40,164),(41,119),(42,166),(43,109),(44,168),(45,111),(46,158),(47,113),(48,160),(49,146),(50,173),(51,148),(52,175),(53,150),(54,177),(55,152),(56,179),(57,154),(58,169),(59,156),(60,171),(62,136),(64,138),(66,140),(68,142),(70,144),(72,134),(73,172),(74,147),(75,174),(76,149),(77,176),(78,151),(79,178),(80,153),(81,180),(82,155),(83,170),(84,145),(85,225),(87,227),(89,217),(91,219),(93,221),(95,223),(97,216),(98,130),(99,206),(100,132),(101,208),(102,122),(103,210),(104,124),(105,212),(106,126),(107,214),(108,128),(110,230),(112,232),(114,234),(116,236),(118,238),(120,240),(133,181),(135,183),(137,185),(139,187),(141,189),(143,191),(157,231),(159,233),(161,235),(163,237),(165,239),(167,229),(194,220),(196,222),(198,224),(200,226),(202,228),(204,218)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 60 | 2 | 4 | 6 | 6 | 10 | 10 | 12 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C5⋊D4 | C4○D12 | S3×D4 | D4⋊2S3 | S3×D5 | Q8⋊2D5 | C2×S3×D5 | D20⋊S3 | C12.28D10 | S3×C5⋊D4 |
| kernel | (C2×Dic6)⋊D5 | Dic3×Dic5 | D10⋊Dic3 | D30⋊4C4 | C3×D10⋊C4 | D30⋊3C4 | C2×C3⋊D20 | C10×Dic6 | D10⋊C4 | C5×Dic3 | C2×Dic6 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | Dic3 | C10 | C10 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 8 | 4 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of (C2×Dic6)⋊D5 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 23 | 38 | 0 | 0 |
| 23 | 46 | 0 | 0 |
| 0 | 0 | 31 | 17 |
| 0 | 0 | 44 | 30 |
| 18 | 9 | 0 | 0 |
| 52 | 43 | 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 | 43 |
| 52 | 43 | 0 | 0 |
| 18 | 9 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[23,23,0,0,38,46,0,0,0,0,31,44,0,0,17,30],[18,52,0,0,9,43,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,43],[52,18,0,0,43,9,0,0,0,0,0,60,0,0,60,0] >;
(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(480,531);
// by ID
G=gap.SmallGroup(480,531);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,219,100,1356,18822]);
// 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=b^-1,b*d=d*b,e*b*e=a*b^7,c*d=d*c,e*c*e=b^6*c,e*d*e=d^-1>;
// generators/relations