metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4.5D5, (C4×Dic15)⋊9C2, (C2×C20).183D6, D6⋊Dic5.6C2, C6.Dic10⋊7C2, C30.Q8⋊8C2, C15⋊9(C42⋊2C2), C30.33(C4○D4), C6.25(C4○D20), C10.D4⋊10S3, C6.7(D4⋊2D5), (C2×C12).182D10, (C2×C30).57C23, (C2×Dic5).15D6, (C22×S3).6D10, C10.29(C4○D12), (C2×C60).160C22, (C2×Dic3).15D10, C3⋊3(C23.D10), C2.11(D12⋊D5), C10.67(D4⋊2S3), C10.26(Q8⋊3S3), (C6×Dic5).34C22, C2.18(D6.D10), C2.13(C30.C23), (C10×Dic3).34C22, (C2×Dic15).187C22, C5⋊6(C4⋊C4⋊S3), (C5×D6⋊C4).5C2, (C2×C4).173(S3×D5), (S3×C2×C10).6C22, C22.144(C2×S3×D5), (C2×C6).69(C22×D5), (C3×C10.D4)⋊10C2, (C2×C10).69(C22×S3), SmallGroup(480,443)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊Dic5.C2
G = < a,b,c,d,e | a6=b2=c10=1, d2=c5, e2=a3c5, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=a3b, ebe-1=bc5, dcd-1=ece-1=c-1, ede-1=a3c5d >
Subgroups: 524 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C4⋊C4⋊S3, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C23.D10, D6⋊Dic5, C30.Q8, C6.Dic10, C3×C10.D4, C5×D6⋊C4, C4×Dic15, D6⋊Dic5.C2
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C42⋊2C2, C22×D5, C4○D12, D4⋊2S3, Q8⋊3S3, S3×D5, C4○D20, D4⋊2D5, C4⋊C4⋊S3, C2×S3×D5, C23.D10, D12⋊D5, D6.D10, C30.C23, D6⋊Dic5.C2
►On 240 points
Generators in S240
(1 62 27 56 49 11)(2 63 28 57 50 12)(3 64 29 58 41 13)(4 65 30 59 42 14)(5 66 21 60 43 15)(6 67 22 51 44 16)(7 68 23 52 45 17)(8 69 24 53 46 18)(9 70 25 54 47 19)(10 61 26 55 48 20)(31 203 239 226 216 195)(32 204 240 227 217 196)(33 205 231 228 218 197)(34 206 232 229 219 198)(35 207 233 230 220 199)(36 208 234 221 211 200)(37 209 235 222 212 191)(38 210 236 223 213 192)(39 201 237 224 214 193)(40 202 238 225 215 194)(71 121 83 112 105 95)(72 122 84 113 106 96)(73 123 85 114 107 97)(74 124 86 115 108 98)(75 125 87 116 109 99)(76 126 88 117 110 100)(77 127 89 118 101 91)(78 128 90 119 102 92)(79 129 81 120 103 93)(80 130 82 111 104 94)(131 184 149 175 162 152)(132 185 150 176 163 153)(133 186 141 177 164 154)(134 187 142 178 165 155)(135 188 143 179 166 156)(136 189 144 180 167 157)(137 190 145 171 168 158)(138 181 146 172 169 159)(139 182 147 173 170 160)(140 183 148 174 161 151)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 51)(28 52)(29 53)(30 54)(31 36)(32 37)(33 38)(34 39)(35 40)(41 69)(42 70)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 81)(80 82)(91 118)(92 119)(93 120)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 151)(141 177)(142 178)(143 179)(144 180)(145 171)(146 172)(147 173)(148 174)(149 175)(150 176)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)(169 181)(170 182)(191 204)(192 205)(193 206)(194 207)(195 208)(196 209)(197 210)(198 201)(199 202)(200 203)(211 239)(212 240)(213 231)(214 232)(215 233)(216 234)(217 235)(218 236)(219 237)(220 238)(221 226)(222 227)(223 228)(224 229)(225 230)
(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)
(1 195 6 200)(2 194 7 199)(3 193 8 198)(4 192 9 197)(5 191 10 196)(11 216 16 211)(12 215 17 220)(13 214 18 219)(14 213 19 218)(15 212 20 217)(21 209 26 204)(22 208 27 203)(23 207 28 202)(24 206 29 201)(25 205 30 210)(31 67 36 62)(32 66 37 61)(33 65 38 70)(34 64 39 69)(35 63 40 68)(41 224 46 229)(42 223 47 228)(43 222 48 227)(44 221 49 226)(45 230 50 225)(51 234 56 239)(52 233 57 238)(53 232 58 237)(54 231 59 236)(55 240 60 235)(71 134 76 139)(72 133 77 138)(73 132 78 137)(74 131 79 136)(75 140 80 135)(81 144 86 149)(82 143 87 148)(83 142 88 147)(84 141 89 146)(85 150 90 145)(91 159 96 154)(92 158 97 153)(93 157 98 152)(94 156 99 151)(95 155 100 160)(101 169 106 164)(102 168 107 163)(103 167 108 162)(104 166 109 161)(105 165 110 170)(111 179 116 174)(112 178 117 173)(113 177 118 172)(114 176 119 171)(115 175 120 180)(121 187 126 182)(122 186 127 181)(123 185 128 190)(124 184 129 189)(125 183 130 188)
(1 179 51 140)(2 178 52 139)(3 177 53 138)(4 176 54 137)(5 175 55 136)(6 174 56 135)(7 173 57 134)(8 172 58 133)(9 171 59 132)(10 180 60 131)(11 143 22 151)(12 142 23 160)(13 141 24 159)(14 150 25 158)(15 149 26 157)(16 148 27 156)(17 147 28 155)(18 146 29 154)(19 145 30 153)(20 144 21 152)(31 130 221 109)(32 129 222 108)(33 128 223 107)(34 127 224 106)(35 126 225 105)(36 125 226 104)(37 124 227 103)(38 123 228 102)(39 122 229 101)(40 121 230 110)(41 186 69 169)(42 185 70 168)(43 184 61 167)(44 183 62 166)(45 182 63 165)(46 181 64 164)(47 190 65 163)(48 189 66 162)(49 188 67 161)(50 187 68 170)(71 233 117 194)(72 232 118 193)(73 231 119 192)(74 240 120 191)(75 239 111 200)(76 238 112 199)(77 237 113 198)(78 236 114 197)(79 235 115 196)(80 234 116 195)(81 212 98 204)(82 211 99 203)(83 220 100 202)(84 219 91 201)(85 218 92 210)(86 217 93 209)(87 216 94 208)(88 215 95 207)(89 214 96 206)(90 213 97 205)
G:=sub<Sym(240)| (1,62,27,56,49,11)(2,63,28,57,50,12)(3,64,29,58,41,13)(4,65,30,59,42,14)(5,66,21,60,43,15)(6,67,22,51,44,16)(7,68,23,52,45,17)(8,69,24,53,46,18)(9,70,25,54,47,19)(10,61,26,55,48,20)(31,203,239,226,216,195)(32,204,240,227,217,196)(33,205,231,228,218,197)(34,206,232,229,219,198)(35,207,233,230,220,199)(36,208,234,221,211,200)(37,209,235,222,212,191)(38,210,236,223,213,192)(39,201,237,224,214,193)(40,202,238,225,215,194)(71,121,83,112,105,95)(72,122,84,113,106,96)(73,123,85,114,107,97)(74,124,86,115,108,98)(75,125,87,116,109,99)(76,126,88,117,110,100)(77,127,89,118,101,91)(78,128,90,119,102,92)(79,129,81,120,103,93)(80,130,82,111,104,94)(131,184,149,175,162,152)(132,185,150,176,163,153)(133,186,141,177,164,154)(134,187,142,178,165,155)(135,188,143,179,166,156)(136,189,144,180,167,157)(137,190,145,171,168,158)(138,181,146,172,169,159)(139,182,147,173,170,160)(140,183,148,174,161,151), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,51)(28,52)(29,53)(30,54)(31,36)(32,37)(33,38)(34,39)(35,40)(41,69)(42,70)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,81)(80,82)(91,118)(92,119)(93,120)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,151)(141,177)(142,178)(143,179)(144,180)(145,171)(146,172)(147,173)(148,174)(149,175)(150,176)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190)(169,181)(170,182)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(197,210)(198,201)(199,202)(200,203)(211,239)(212,240)(213,231)(214,232)(215,233)(216,234)(217,235)(218,236)(219,237)(220,238)(221,226)(222,227)(223,228)(224,229)(225,230), (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), (1,195,6,200)(2,194,7,199)(3,193,8,198)(4,192,9,197)(5,191,10,196)(11,216,16,211)(12,215,17,220)(13,214,18,219)(14,213,19,218)(15,212,20,217)(21,209,26,204)(22,208,27,203)(23,207,28,202)(24,206,29,201)(25,205,30,210)(31,67,36,62)(32,66,37,61)(33,65,38,70)(34,64,39,69)(35,63,40,68)(41,224,46,229)(42,223,47,228)(43,222,48,227)(44,221,49,226)(45,230,50,225)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,134,76,139)(72,133,77,138)(73,132,78,137)(74,131,79,136)(75,140,80,135)(81,144,86,149)(82,143,87,148)(83,142,88,147)(84,141,89,146)(85,150,90,145)(91,159,96,154)(92,158,97,153)(93,157,98,152)(94,156,99,151)(95,155,100,160)(101,169,106,164)(102,168,107,163)(103,167,108,162)(104,166,109,161)(105,165,110,170)(111,179,116,174)(112,178,117,173)(113,177,118,172)(114,176,119,171)(115,175,120,180)(121,187,126,182)(122,186,127,181)(123,185,128,190)(124,184,129,189)(125,183,130,188), (1,179,51,140)(2,178,52,139)(3,177,53,138)(4,176,54,137)(5,175,55,136)(6,174,56,135)(7,173,57,134)(8,172,58,133)(9,171,59,132)(10,180,60,131)(11,143,22,151)(12,142,23,160)(13,141,24,159)(14,150,25,158)(15,149,26,157)(16,148,27,156)(17,147,28,155)(18,146,29,154)(19,145,30,153)(20,144,21,152)(31,130,221,109)(32,129,222,108)(33,128,223,107)(34,127,224,106)(35,126,225,105)(36,125,226,104)(37,124,227,103)(38,123,228,102)(39,122,229,101)(40,121,230,110)(41,186,69,169)(42,185,70,168)(43,184,61,167)(44,183,62,166)(45,182,63,165)(46,181,64,164)(47,190,65,163)(48,189,66,162)(49,188,67,161)(50,187,68,170)(71,233,117,194)(72,232,118,193)(73,231,119,192)(74,240,120,191)(75,239,111,200)(76,238,112,199)(77,237,113,198)(78,236,114,197)(79,235,115,196)(80,234,116,195)(81,212,98,204)(82,211,99,203)(83,220,100,202)(84,219,91,201)(85,218,92,210)(86,217,93,209)(87,216,94,208)(88,215,95,207)(89,214,96,206)(90,213,97,205)>;
G:=Group( (1,62,27,56,49,11)(2,63,28,57,50,12)(3,64,29,58,41,13)(4,65,30,59,42,14)(5,66,21,60,43,15)(6,67,22,51,44,16)(7,68,23,52,45,17)(8,69,24,53,46,18)(9,70,25,54,47,19)(10,61,26,55,48,20)(31,203,239,226,216,195)(32,204,240,227,217,196)(33,205,231,228,218,197)(34,206,232,229,219,198)(35,207,233,230,220,199)(36,208,234,221,211,200)(37,209,235,222,212,191)(38,210,236,223,213,192)(39,201,237,224,214,193)(40,202,238,225,215,194)(71,121,83,112,105,95)(72,122,84,113,106,96)(73,123,85,114,107,97)(74,124,86,115,108,98)(75,125,87,116,109,99)(76,126,88,117,110,100)(77,127,89,118,101,91)(78,128,90,119,102,92)(79,129,81,120,103,93)(80,130,82,111,104,94)(131,184,149,175,162,152)(132,185,150,176,163,153)(133,186,141,177,164,154)(134,187,142,178,165,155)(135,188,143,179,166,156)(136,189,144,180,167,157)(137,190,145,171,168,158)(138,181,146,172,169,159)(139,182,147,173,170,160)(140,183,148,174,161,151), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,51)(28,52)(29,53)(30,54)(31,36)(32,37)(33,38)(34,39)(35,40)(41,69)(42,70)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,81)(80,82)(91,118)(92,119)(93,120)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,151)(141,177)(142,178)(143,179)(144,180)(145,171)(146,172)(147,173)(148,174)(149,175)(150,176)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190)(169,181)(170,182)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(197,210)(198,201)(199,202)(200,203)(211,239)(212,240)(213,231)(214,232)(215,233)(216,234)(217,235)(218,236)(219,237)(220,238)(221,226)(222,227)(223,228)(224,229)(225,230), (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), (1,195,6,200)(2,194,7,199)(3,193,8,198)(4,192,9,197)(5,191,10,196)(11,216,16,211)(12,215,17,220)(13,214,18,219)(14,213,19,218)(15,212,20,217)(21,209,26,204)(22,208,27,203)(23,207,28,202)(24,206,29,201)(25,205,30,210)(31,67,36,62)(32,66,37,61)(33,65,38,70)(34,64,39,69)(35,63,40,68)(41,224,46,229)(42,223,47,228)(43,222,48,227)(44,221,49,226)(45,230,50,225)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,134,76,139)(72,133,77,138)(73,132,78,137)(74,131,79,136)(75,140,80,135)(81,144,86,149)(82,143,87,148)(83,142,88,147)(84,141,89,146)(85,150,90,145)(91,159,96,154)(92,158,97,153)(93,157,98,152)(94,156,99,151)(95,155,100,160)(101,169,106,164)(102,168,107,163)(103,167,108,162)(104,166,109,161)(105,165,110,170)(111,179,116,174)(112,178,117,173)(113,177,118,172)(114,176,119,171)(115,175,120,180)(121,187,126,182)(122,186,127,181)(123,185,128,190)(124,184,129,189)(125,183,130,188), (1,179,51,140)(2,178,52,139)(3,177,53,138)(4,176,54,137)(5,175,55,136)(6,174,56,135)(7,173,57,134)(8,172,58,133)(9,171,59,132)(10,180,60,131)(11,143,22,151)(12,142,23,160)(13,141,24,159)(14,150,25,158)(15,149,26,157)(16,148,27,156)(17,147,28,155)(18,146,29,154)(19,145,30,153)(20,144,21,152)(31,130,221,109)(32,129,222,108)(33,128,223,107)(34,127,224,106)(35,126,225,105)(36,125,226,104)(37,124,227,103)(38,123,228,102)(39,122,229,101)(40,121,230,110)(41,186,69,169)(42,185,70,168)(43,184,61,167)(44,183,62,166)(45,182,63,165)(46,181,64,164)(47,190,65,163)(48,189,66,162)(49,188,67,161)(50,187,68,170)(71,233,117,194)(72,232,118,193)(73,231,119,192)(74,240,120,191)(75,239,111,200)(76,238,112,199)(77,237,113,198)(78,236,114,197)(79,235,115,196)(80,234,116,195)(81,212,98,204)(82,211,99,203)(83,220,100,202)(84,219,91,201)(85,218,92,210)(86,217,93,209)(87,216,94,208)(88,215,95,207)(89,214,96,206)(90,213,97,205) );
G=PermutationGroup([[(1,62,27,56,49,11),(2,63,28,57,50,12),(3,64,29,58,41,13),(4,65,30,59,42,14),(5,66,21,60,43,15),(6,67,22,51,44,16),(7,68,23,52,45,17),(8,69,24,53,46,18),(9,70,25,54,47,19),(10,61,26,55,48,20),(31,203,239,226,216,195),(32,204,240,227,217,196),(33,205,231,228,218,197),(34,206,232,229,219,198),(35,207,233,230,220,199),(36,208,234,221,211,200),(37,209,235,222,212,191),(38,210,236,223,213,192),(39,201,237,224,214,193),(40,202,238,225,215,194),(71,121,83,112,105,95),(72,122,84,113,106,96),(73,123,85,114,107,97),(74,124,86,115,108,98),(75,125,87,116,109,99),(76,126,88,117,110,100),(77,127,89,118,101,91),(78,128,90,119,102,92),(79,129,81,120,103,93),(80,130,82,111,104,94),(131,184,149,175,162,152),(132,185,150,176,163,153),(133,186,141,177,164,154),(134,187,142,178,165,155),(135,188,143,179,166,156),(136,189,144,180,167,157),(137,190,145,171,168,158),(138,181,146,172,169,159),(139,182,147,173,170,160),(140,183,148,174,161,151)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,51),(28,52),(29,53),(30,54),(31,36),(32,37),(33,38),(34,39),(35,40),(41,69),(42,70),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(71,83),(72,84),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,81),(80,82),(91,118),(92,119),(93,120),(94,111),(95,112),(96,113),(97,114),(98,115),(99,116),(100,117),(131,152),(132,153),(133,154),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,151),(141,177),(142,178),(143,179),(144,180),(145,171),(146,172),(147,173),(148,174),(149,175),(150,176),(161,183),(162,184),(163,185),(164,186),(165,187),(166,188),(167,189),(168,190),(169,181),(170,182),(191,204),(192,205),(193,206),(194,207),(195,208),(196,209),(197,210),(198,201),(199,202),(200,203),(211,239),(212,240),(213,231),(214,232),(215,233),(216,234),(217,235),(218,236),(219,237),(220,238),(221,226),(222,227),(223,228),(224,229),(225,230)], [(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)], [(1,195,6,200),(2,194,7,199),(3,193,8,198),(4,192,9,197),(5,191,10,196),(11,216,16,211),(12,215,17,220),(13,214,18,219),(14,213,19,218),(15,212,20,217),(21,209,26,204),(22,208,27,203),(23,207,28,202),(24,206,29,201),(25,205,30,210),(31,67,36,62),(32,66,37,61),(33,65,38,70),(34,64,39,69),(35,63,40,68),(41,224,46,229),(42,223,47,228),(43,222,48,227),(44,221,49,226),(45,230,50,225),(51,234,56,239),(52,233,57,238),(53,232,58,237),(54,231,59,236),(55,240,60,235),(71,134,76,139),(72,133,77,138),(73,132,78,137),(74,131,79,136),(75,140,80,135),(81,144,86,149),(82,143,87,148),(83,142,88,147),(84,141,89,146),(85,150,90,145),(91,159,96,154),(92,158,97,153),(93,157,98,152),(94,156,99,151),(95,155,100,160),(101,169,106,164),(102,168,107,163),(103,167,108,162),(104,166,109,161),(105,165,110,170),(111,179,116,174),(112,178,117,173),(113,177,118,172),(114,176,119,171),(115,175,120,180),(121,187,126,182),(122,186,127,181),(123,185,128,190),(124,184,129,189),(125,183,130,188)], [(1,179,51,140),(2,178,52,139),(3,177,53,138),(4,176,54,137),(5,175,55,136),(6,174,56,135),(7,173,57,134),(8,172,58,133),(9,171,59,132),(10,180,60,131),(11,143,22,151),(12,142,23,160),(13,141,24,159),(14,150,25,158),(15,149,26,157),(16,148,27,156),(17,147,28,155),(18,146,29,154),(19,145,30,153),(20,144,21,152),(31,130,221,109),(32,129,222,108),(33,128,223,107),(34,127,224,106),(35,126,225,105),(36,125,226,104),(37,124,227,103),(38,123,228,102),(39,122,229,101),(40,121,230,110),(41,186,69,169),(42,185,70,168),(43,184,61,167),(44,183,62,166),(45,182,63,165),(46,181,64,164),(47,190,65,163),(48,189,66,162),(49,188,67,161),(50,187,68,170),(71,233,117,194),(72,232,118,193),(73,231,119,192),(74,240,120,191),(75,239,111,200),(76,238,112,199),(77,237,113,198),(78,236,114,197),(79,235,115,196),(80,234,116,195),(81,212,98,204),(82,211,99,203),(83,220,100,202),(84,219,91,201),(85,218,92,210),(86,217,93,209),(87,216,94,208),(88,215,95,207),(89,214,96,206),(90,213,97,205)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 12 | 20 | 20 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | C4○D20 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | C2×S3×D5 | D12⋊D5 | D6.D10 | C30.C23 |
| kernel | D6⋊Dic5.C2 | D6⋊Dic5 | C30.Q8 | C6.Dic10 | C3×C10.D4 | C5×D6⋊C4 | C4×Dic15 | C10.D4 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 6 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 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 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 50 | 41 | 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 |
| 13 | 5 | 0 | 0 | 0 | 0 |
| 27 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 22 | 0 | 0 |
| 0 | 0 | 50 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 21 | 55 | 0 | 0 | 0 | 0 |
| 53 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 50 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[1,7,0,0,0,0,0,60,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,1],[3,50,0,0,0,0,0,41,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,27,0,0,0,0,5,48,0,0,0,0,0,0,11,50,0,0,0,0,22,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[21,53,0,0,0,0,55,40,0,0,0,0,0,0,11,50,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D6⋊Dic5.C2 in GAP, Magma, Sage, TeX
D_6\rtimes {\rm Dic}_5.C_2 % in TeX
G:=Group("D6:Dic5.C2"); // GroupNames label
G:=SmallGroup(480,443);
// by ID
G=gap.SmallGroup(480,443);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^10=1,d^2=c^5,e^2=a^3*c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=a^3*b,e*b*e^-1=b*c^5,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=a^3*c^5*d>;
// generators/relations