metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊Dic5⋊3C2, D30⋊4C4⋊4C2, (S3×C10).28D4, C30.104(C2×D4), (C2×C20).256D6, C10.126(S3×D4), C10.D4⋊1S3, C5⋊6(D6.D4), Dic15⋊5C4⋊3C2, (C2×Dic5).7D6, D30⋊3C4⋊29C2, C30.18(C4○D4), C6.18(C4○D20), (C2×C12).175D10, D6.11(C5⋊D4), (C2×C30).41C23, C10.21(C4○D12), (C2×C60).400C22, C10.7(Q8⋊3S3), (C22×S3).64D10, C15⋊2(C22.D4), C2.10(D60⋊C2), (C2×Dic3).134D10, C3⋊1(C23.23D10), (C6×Dic5).23C22, C2.11(D6.D10), (C2×Dic15).46C22, (C22×D15).20C22, (C10×Dic3).158C22, (S3×C2×C4)⋊7D5, (S3×C2×C20)⋊17C2, C2.9(S3×C5⋊D4), (C2×C4).68(S3×D5), C6.27(C2×C5⋊D4), (C2×C5⋊D12).3C2, C22.130(C2×S3×D5), (S3×C2×C10).76C22, (C2×C6).53(C22×D5), (C3×C10.D4)⋊28C2, (C2×C10).53(C22×S3), SmallGroup(480,427)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊Dic5⋊C2
G = < a,b,c,d,e | a6=b2=c10=e2=1, d2=c5, bab=eae=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, ebe=ab, dcd-1=ece=c-1, ede=a3c5d >
Subgroups: 844 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10, D30, C2×C30, C10.D4, C10.D4, D10⋊C4, C23.D5, C2×C5⋊D4, C22×C20, D6.D4, C5⋊D12, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C23.23D10, D6⋊Dic5, D30⋊4C4, Dic15⋊5C4, C3×C10.D4, D30⋊3C4, C2×C5⋊D12, S3×C2×C20, D6⋊Dic5⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C22.D4, C5⋊D4, C22×D5, C4○D12, S3×D4, Q8⋊3S3, S3×D5, C4○D20, C2×C5⋊D4, D6.D4, C2×S3×D5, C23.23D10, D60⋊C2, D6.D10, S3×C5⋊D4, D6⋊Dic5⋊C2
►On 240 points
Generators in S240
(1 62 21 51 32 47)(2 63 22 52 33 48)(3 64 23 53 34 49)(4 65 24 54 35 50)(5 66 25 55 36 41)(6 67 26 56 37 42)(7 68 27 57 38 43)(8 69 28 58 39 44)(9 70 29 59 40 45)(10 61 30 60 31 46)(11 193 234 218 223 204)(12 194 235 219 224 205)(13 195 236 220 225 206)(14 196 237 211 226 207)(15 197 238 212 227 208)(16 198 239 213 228 209)(17 199 240 214 229 210)(18 200 231 215 230 201)(19 191 232 216 221 202)(20 192 233 217 222 203)(71 115 96 101 85 129)(72 116 97 102 86 130)(73 117 98 103 87 121)(74 118 99 104 88 122)(75 119 100 105 89 123)(76 120 91 106 90 124)(77 111 92 107 81 125)(78 112 93 108 82 126)(79 113 94 109 83 127)(80 114 95 110 84 128)(131 172 156 161 142 186)(132 173 157 162 143 187)(133 174 158 163 144 188)(134 175 159 164 145 189)(135 176 160 165 146 190)(136 177 151 166 147 181)(137 178 152 167 148 182)(138 179 153 168 149 183)(139 180 154 169 150 184)(140 171 155 170 141 185)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 41)(11 16)(12 17)(13 18)(14 19)(15 20)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 55)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(71 120)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 101)(91 129)(92 130)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)(151 156)(152 157)(153 158)(154 159)(155 160)(161 177)(162 178)(163 179)(164 180)(165 171)(166 172)(167 173)(168 174)(169 175)(170 176)(181 186)(182 187)(183 188)(184 189)(185 190)(191 207)(192 208)(193 209)(194 210)(195 201)(196 202)(197 203)(198 204)(199 205)(200 206)(211 216)(212 217)(213 218)(214 219)(215 220)(221 237)(222 238)(223 239)(224 240)(225 231)(226 232)(227 233)(228 234)(229 235)(230 236)
(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)
(1 144 6 149)(2 143 7 148)(3 142 8 147)(4 141 9 146)(5 150 10 145)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 133 26 138)(22 132 27 137)(23 131 28 136)(24 140 29 135)(25 139 30 134)(31 159 36 154)(32 158 37 153)(33 157 38 152)(34 156 39 151)(35 155 40 160)(41 169 46 164)(42 168 47 163)(43 167 48 162)(44 166 49 161)(45 165 50 170)(51 174 56 179)(52 173 57 178)(53 172 58 177)(54 171 59 176)(55 180 60 175)(61 189 66 184)(62 188 67 183)(63 187 68 182)(64 186 69 181)(65 185 70 190)(71 199 76 194)(72 198 77 193)(73 197 78 192)(74 196 79 191)(75 195 80 200)(81 204 86 209)(82 203 87 208)(83 202 88 207)(84 201 89 206)(85 210 90 205)(91 219 96 214)(92 218 97 213)(93 217 98 212)(94 216 99 211)(95 215 100 220)(101 229 106 224)(102 228 107 223)(103 227 108 222)(104 226 109 221)(105 225 110 230)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)
(1 209)(2 208)(3 207)(4 206)(5 205)(6 204)(7 203)(8 202)(9 201)(10 210)(11 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 213)(22 212)(23 211)(24 220)(25 219)(26 218)(27 217)(28 216)(29 215)(30 214)(31 199)(32 198)(33 197)(34 196)(35 195)(36 194)(37 193)(38 192)(39 191)(40 200)(51 239)(52 238)(53 237)(54 236)(55 235)(56 234)(57 233)(58 232)(59 231)(60 240)(61 229)(62 228)(63 227)(64 226)(65 225)(66 224)(67 223)(68 222)(69 221)(70 230)(71 189)(72 188)(73 187)(74 186)(75 185)(76 184)(77 183)(78 182)(79 181)(80 190)(81 179)(82 178)(83 177)(84 176)(85 175)(86 174)(87 173)(88 172)(89 171)(90 180)(91 169)(92 168)(93 167)(94 166)(95 165)(96 164)(97 163)(98 162)(99 161)(100 170)(101 159)(102 158)(103 157)(104 156)(105 155)(106 154)(107 153)(108 152)(109 151)(110 160)(111 149)(112 148)(113 147)(114 146)(115 145)(116 144)(117 143)(118 142)(119 141)(120 150)(121 132)(122 131)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)
G:=sub<Sym(240)| (1,62,21,51,32,47)(2,63,22,52,33,48)(3,64,23,53,34,49)(4,65,24,54,35,50)(5,66,25,55,36,41)(6,67,26,56,37,42)(7,68,27,57,38,43)(8,69,28,58,39,44)(9,70,29,59,40,45)(10,61,30,60,31,46)(11,193,234,218,223,204)(12,194,235,219,224,205)(13,195,236,220,225,206)(14,196,237,211,226,207)(15,197,238,212,227,208)(16,198,239,213,228,209)(17,199,240,214,229,210)(18,200,231,215,230,201)(19,191,232,216,221,202)(20,192,233,217,222,203)(71,115,96,101,85,129)(72,116,97,102,86,130)(73,117,98,103,87,121)(74,118,99,104,88,122)(75,119,100,105,89,123)(76,120,91,106,90,124)(77,111,92,107,81,125)(78,112,93,108,82,126)(79,113,94,109,83,127)(80,114,95,110,84,128)(131,172,156,161,142,186)(132,173,157,162,143,187)(133,174,158,163,144,188)(134,175,159,164,145,189)(135,176,160,165,146,190)(136,177,151,166,147,181)(137,178,152,167,148,182)(138,179,153,168,149,183)(139,180,154,169,150,184)(140,171,155,170,141,185), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,41)(11,16)(12,17)(13,18)(14,19)(15,20)(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(71,120)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,101)(91,129)(92,130)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(151,156)(152,157)(153,158)(154,159)(155,160)(161,177)(162,178)(163,179)(164,180)(165,171)(166,172)(167,173)(168,174)(169,175)(170,176)(181,186)(182,187)(183,188)(184,189)(185,190)(191,207)(192,208)(193,209)(194,210)(195,201)(196,202)(197,203)(198,204)(199,205)(200,206)(211,216)(212,217)(213,218)(214,219)(215,220)(221,237)(222,238)(223,239)(224,240)(225,231)(226,232)(227,233)(228,234)(229,235)(230,236), (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), (1,144,6,149)(2,143,7,148)(3,142,8,147)(4,141,9,146)(5,150,10,145)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,133,26,138)(22,132,27,137)(23,131,28,136)(24,140,29,135)(25,139,30,134)(31,159,36,154)(32,158,37,153)(33,157,38,152)(34,156,39,151)(35,155,40,160)(41,169,46,164)(42,168,47,163)(43,167,48,162)(44,166,49,161)(45,165,50,170)(51,174,56,179)(52,173,57,178)(53,172,58,177)(54,171,59,176)(55,180,60,175)(61,189,66,184)(62,188,67,183)(63,187,68,182)(64,186,69,181)(65,185,70,190)(71,199,76,194)(72,198,77,193)(73,197,78,192)(74,196,79,191)(75,195,80,200)(81,204,86,209)(82,203,87,208)(83,202,88,207)(84,201,89,206)(85,210,90,205)(91,219,96,214)(92,218,97,213)(93,217,98,212)(94,216,99,211)(95,215,100,220)(101,229,106,224)(102,228,107,223)(103,227,108,222)(104,226,109,221)(105,225,110,230)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235), (1,209)(2,208)(3,207)(4,206)(5,205)(6,204)(7,203)(8,202)(9,201)(10,210)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,213)(22,212)(23,211)(24,220)(25,219)(26,218)(27,217)(28,216)(29,215)(30,214)(31,199)(32,198)(33,197)(34,196)(35,195)(36,194)(37,193)(38,192)(39,191)(40,200)(51,239)(52,238)(53,237)(54,236)(55,235)(56,234)(57,233)(58,232)(59,231)(60,240)(61,229)(62,228)(63,227)(64,226)(65,225)(66,224)(67,223)(68,222)(69,221)(70,230)(71,189)(72,188)(73,187)(74,186)(75,185)(76,184)(77,183)(78,182)(79,181)(80,190)(81,179)(82,178)(83,177)(84,176)(85,175)(86,174)(87,173)(88,172)(89,171)(90,180)(91,169)(92,168)(93,167)(94,166)(95,165)(96,164)(97,163)(98,162)(99,161)(100,170)(101,159)(102,158)(103,157)(104,156)(105,155)(106,154)(107,153)(108,152)(109,151)(110,160)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,150)(121,132)(122,131)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)>;
G:=Group( (1,62,21,51,32,47)(2,63,22,52,33,48)(3,64,23,53,34,49)(4,65,24,54,35,50)(5,66,25,55,36,41)(6,67,26,56,37,42)(7,68,27,57,38,43)(8,69,28,58,39,44)(9,70,29,59,40,45)(10,61,30,60,31,46)(11,193,234,218,223,204)(12,194,235,219,224,205)(13,195,236,220,225,206)(14,196,237,211,226,207)(15,197,238,212,227,208)(16,198,239,213,228,209)(17,199,240,214,229,210)(18,200,231,215,230,201)(19,191,232,216,221,202)(20,192,233,217,222,203)(71,115,96,101,85,129)(72,116,97,102,86,130)(73,117,98,103,87,121)(74,118,99,104,88,122)(75,119,100,105,89,123)(76,120,91,106,90,124)(77,111,92,107,81,125)(78,112,93,108,82,126)(79,113,94,109,83,127)(80,114,95,110,84,128)(131,172,156,161,142,186)(132,173,157,162,143,187)(133,174,158,163,144,188)(134,175,159,164,145,189)(135,176,160,165,146,190)(136,177,151,166,147,181)(137,178,152,167,148,182)(138,179,153,168,149,183)(139,180,154,169,150,184)(140,171,155,170,141,185), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,41)(11,16)(12,17)(13,18)(14,19)(15,20)(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(71,120)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,101)(91,129)(92,130)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(151,156)(152,157)(153,158)(154,159)(155,160)(161,177)(162,178)(163,179)(164,180)(165,171)(166,172)(167,173)(168,174)(169,175)(170,176)(181,186)(182,187)(183,188)(184,189)(185,190)(191,207)(192,208)(193,209)(194,210)(195,201)(196,202)(197,203)(198,204)(199,205)(200,206)(211,216)(212,217)(213,218)(214,219)(215,220)(221,237)(222,238)(223,239)(224,240)(225,231)(226,232)(227,233)(228,234)(229,235)(230,236), (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), (1,144,6,149)(2,143,7,148)(3,142,8,147)(4,141,9,146)(5,150,10,145)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,133,26,138)(22,132,27,137)(23,131,28,136)(24,140,29,135)(25,139,30,134)(31,159,36,154)(32,158,37,153)(33,157,38,152)(34,156,39,151)(35,155,40,160)(41,169,46,164)(42,168,47,163)(43,167,48,162)(44,166,49,161)(45,165,50,170)(51,174,56,179)(52,173,57,178)(53,172,58,177)(54,171,59,176)(55,180,60,175)(61,189,66,184)(62,188,67,183)(63,187,68,182)(64,186,69,181)(65,185,70,190)(71,199,76,194)(72,198,77,193)(73,197,78,192)(74,196,79,191)(75,195,80,200)(81,204,86,209)(82,203,87,208)(83,202,88,207)(84,201,89,206)(85,210,90,205)(91,219,96,214)(92,218,97,213)(93,217,98,212)(94,216,99,211)(95,215,100,220)(101,229,106,224)(102,228,107,223)(103,227,108,222)(104,226,109,221)(105,225,110,230)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235), (1,209)(2,208)(3,207)(4,206)(5,205)(6,204)(7,203)(8,202)(9,201)(10,210)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,213)(22,212)(23,211)(24,220)(25,219)(26,218)(27,217)(28,216)(29,215)(30,214)(31,199)(32,198)(33,197)(34,196)(35,195)(36,194)(37,193)(38,192)(39,191)(40,200)(51,239)(52,238)(53,237)(54,236)(55,235)(56,234)(57,233)(58,232)(59,231)(60,240)(61,229)(62,228)(63,227)(64,226)(65,225)(66,224)(67,223)(68,222)(69,221)(70,230)(71,189)(72,188)(73,187)(74,186)(75,185)(76,184)(77,183)(78,182)(79,181)(80,190)(81,179)(82,178)(83,177)(84,176)(85,175)(86,174)(87,173)(88,172)(89,171)(90,180)(91,169)(92,168)(93,167)(94,166)(95,165)(96,164)(97,163)(98,162)(99,161)(100,170)(101,159)(102,158)(103,157)(104,156)(105,155)(106,154)(107,153)(108,152)(109,151)(110,160)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,150)(121,132)(122,131)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133) );
G=PermutationGroup([[(1,62,21,51,32,47),(2,63,22,52,33,48),(3,64,23,53,34,49),(4,65,24,54,35,50),(5,66,25,55,36,41),(6,67,26,56,37,42),(7,68,27,57,38,43),(8,69,28,58,39,44),(9,70,29,59,40,45),(10,61,30,60,31,46),(11,193,234,218,223,204),(12,194,235,219,224,205),(13,195,236,220,225,206),(14,196,237,211,226,207),(15,197,238,212,227,208),(16,198,239,213,228,209),(17,199,240,214,229,210),(18,200,231,215,230,201),(19,191,232,216,221,202),(20,192,233,217,222,203),(71,115,96,101,85,129),(72,116,97,102,86,130),(73,117,98,103,87,121),(74,118,99,104,88,122),(75,119,100,105,89,123),(76,120,91,106,90,124),(77,111,92,107,81,125),(78,112,93,108,82,126),(79,113,94,109,83,127),(80,114,95,110,84,128),(131,172,156,161,142,186),(132,173,157,162,143,187),(133,174,158,163,144,188),(134,175,159,164,145,189),(135,176,160,165,146,190),(136,177,151,166,147,181),(137,178,152,167,148,182),(138,179,153,168,149,183),(139,180,154,169,150,184),(140,171,155,170,141,185)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,41),(11,16),(12,17),(13,18),(14,19),(15,20),(21,56),(22,57),(23,58),(24,59),(25,60),(26,51),(27,52),(28,53),(29,54),(30,55),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65),(71,120),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,102),(82,103),(83,104),(84,105),(85,106),(86,107),(87,108),(88,109),(89,110),(90,101),(91,129),(92,130),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146),(151,156),(152,157),(153,158),(154,159),(155,160),(161,177),(162,178),(163,179),(164,180),(165,171),(166,172),(167,173),(168,174),(169,175),(170,176),(181,186),(182,187),(183,188),(184,189),(185,190),(191,207),(192,208),(193,209),(194,210),(195,201),(196,202),(197,203),(198,204),(199,205),(200,206),(211,216),(212,217),(213,218),(214,219),(215,220),(221,237),(222,238),(223,239),(224,240),(225,231),(226,232),(227,233),(228,234),(229,235),(230,236)], [(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)], [(1,144,6,149),(2,143,7,148),(3,142,8,147),(4,141,9,146),(5,150,10,145),(11,130,16,125),(12,129,17,124),(13,128,18,123),(14,127,19,122),(15,126,20,121),(21,133,26,138),(22,132,27,137),(23,131,28,136),(24,140,29,135),(25,139,30,134),(31,159,36,154),(32,158,37,153),(33,157,38,152),(34,156,39,151),(35,155,40,160),(41,169,46,164),(42,168,47,163),(43,167,48,162),(44,166,49,161),(45,165,50,170),(51,174,56,179),(52,173,57,178),(53,172,58,177),(54,171,59,176),(55,180,60,175),(61,189,66,184),(62,188,67,183),(63,187,68,182),(64,186,69,181),(65,185,70,190),(71,199,76,194),(72,198,77,193),(73,197,78,192),(74,196,79,191),(75,195,80,200),(81,204,86,209),(82,203,87,208),(83,202,88,207),(84,201,89,206),(85,210,90,205),(91,219,96,214),(92,218,97,213),(93,217,98,212),(94,216,99,211),(95,215,100,220),(101,229,106,224),(102,228,107,223),(103,227,108,222),(104,226,109,221),(105,225,110,230),(111,234,116,239),(112,233,117,238),(113,232,118,237),(114,231,119,236),(115,240,120,235)], [(1,209),(2,208),(3,207),(4,206),(5,205),(6,204),(7,203),(8,202),(9,201),(10,210),(11,42),(12,41),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,213),(22,212),(23,211),(24,220),(25,219),(26,218),(27,217),(28,216),(29,215),(30,214),(31,199),(32,198),(33,197),(34,196),(35,195),(36,194),(37,193),(38,192),(39,191),(40,200),(51,239),(52,238),(53,237),(54,236),(55,235),(56,234),(57,233),(58,232),(59,231),(60,240),(61,229),(62,228),(63,227),(64,226),(65,225),(66,224),(67,223),(68,222),(69,221),(70,230),(71,189),(72,188),(73,187),(74,186),(75,185),(76,184),(77,183),(78,182),(79,181),(80,190),(81,179),(82,178),(83,177),(84,176),(85,175),(86,174),(87,173),(88,172),(89,171),(90,180),(91,169),(92,168),(93,167),(94,166),(95,165),(96,164),(97,163),(98,162),(99,161),(100,170),(101,159),(102,158),(103,157),(104,156),(105,155),(106,154),(107,153),(108,152),(109,151),(110,160),(111,149),(112,148),(113,147),(114,146),(115,145),(116,144),(117,143),(118,142),(119,141),(120,150),(121,132),(122,131),(123,140),(124,139),(125,138),(126,137),(127,136),(128,135),(129,134),(130,133)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 60 | 2 | 2 | 2 | 6 | 6 | 20 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C4○D12 | C4○D20 | S3×D4 | Q8⋊3S3 | S3×D5 | C2×S3×D5 | D60⋊C2 | D6.D10 | S3×C5⋊D4 |
| kernel | D6⋊Dic5⋊C2 | D6⋊Dic5 | D30⋊4C4 | Dic15⋊5C4 | C3×C10.D4 | D30⋊3C4 | C2×C5⋊D12 | S3×C2×C20 | C10.D4 | S3×C10 | S3×C2×C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C10 | C6 | C10 | C10 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 16 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D6⋊Dic5⋊C2 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 42 | 0 | 0 |
| 0 | 0 | 13 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 39 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 42 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 30 | 60 |
| 34 | 0 | 0 | 0 | 0 | 0 |
| 31 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 40 | 41 |
| 34 | 8 | 0 | 0 | 0 | 0 |
| 31 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 3 |
| 0 | 0 | 0 | 0 | 16 | 45 |
| 8 | 27 | 0 | 0 | 0 | 0 |
| 18 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 33 |
| 0 | 0 | 0 | 0 | 54 | 7 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,13,0,0,0,0,42,59,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,39,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,42,60,0,0,0,0,0,0,1,30,0,0,0,0,0,60],[34,31,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,40,0,0,0,0,0,41],[34,31,0,0,0,0,8,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,3,45],[8,18,0,0,0,0,27,53,0,0,0,0,0,0,1,13,0,0,0,0,0,60,0,0,0,0,0,0,54,54,0,0,0,0,33,7] >;
D6⋊Dic5⋊C2 in GAP, Magma, Sage, TeX
D_6\rtimes {\rm Dic}_5\rtimes C_2 % in TeX
G:=Group("D6:Dic5:C2"); // GroupNames label
G:=SmallGroup(480,427);
// by ID
G=gap.SmallGroup(480,427);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^10=e^2=1,d^2=c^5,b*a*b=e*a*e=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,e*b*e=a*b,d*c*d^-1=e*c*e=c^-1,e*d*e=a^3*c^5*d>;
// generators/relations