G = C6.(D4×D5) order 480 = 25·3·5
32nd non-split extension by C6 of D4×D5 acting via D4×D5/C22×D5=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.154(D4×D5), C6.D4⋊5D5, C30.214(C2×D4), C23.16(S3×D5), C6.79(C4○D20), C15⋊15(C4.4D4), C5⋊2(C23.12D6), (C3×Dic5).19D4, (C22×D5).27D6, (C22×C6).24D10, (C22×C10).39D6, (Dic3×Dic5)⋊32C2, C30.136(C4○D4), C6.78(D4⋊2D5), C30.38D4⋊17C2, D10⋊Dic3⋊26C2, (C2×C30).176C23, (C2×Dic5).127D6, C3⋊7(Dic5.5D4), C10.51(D4⋊2S3), (C2×Dic3).119D10, Dic5.15(C3⋊D4), (C22×C30).38C22, C2.23(C30.C23), C2.24(Dic5.D6), (C6×Dic5).104C22, (C10×Dic3).104C22, (C2×Dic15).125C22, (C2×C15⋊Q8)⋊15C2, (C6×C5⋊D4).2C2, (C2×C5⋊D4).2S3, C2.37(D5×C3⋊D4), C10.58(C2×C3⋊D4), (D5×C2×C6).45C22, C22.220(C2×S3×D5), (C5×C6.D4)⋊5C2, (C2×C6).188(C22×D5), (C2×C10).188(C22×S3), SmallGroup(480,610)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.(D4×D5)
G = < a,b,c,d,e | a6=b4=d5=1, c2=e2=a3, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a3b-1, bd=db, be=eb, cd=dc, ece-1=a3c, ede-1=d-1 >
Subgroups: 732 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, D4, Q8, C23, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4.4D4, Dic10, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic3, C6.D4, C6.D4, C2×Dic6, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C2×C30, C2×C30, C4×Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C23.12D6, C15⋊Q8, C6×Dic5, C3×C5⋊D4, C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, Dic5.5D4, Dic3×Dic5, D10⋊Dic3, C5×C6.D4, C30.38D4, C2×C15⋊Q8, C6×C5⋊D4, C6.(D4×D5)
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4.4D4, C22×D5, D4⋊2S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, D4⋊2D5, C23.12D6, C2×S3×D5, Dic5.5D4, Dic5.D6, C30.C23, D5×C3⋊D4, C6.(D4×D5)
►On 240 points
Generators in S240
(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)
(1 163 17 175)(2 168 18 180)(3 167 13 179)(4 166 14 178)(5 165 15 177)(6 164 16 176)(7 111 232 153)(8 110 233 152)(9 109 234 151)(10 114 229 156)(11 113 230 155)(12 112 231 154)(19 125 79 202)(20 124 80 201)(21 123 81 200)(22 122 82 199)(23 121 83 204)(24 126 84 203)(25 134 85 206)(26 133 86 205)(27 138 87 210)(28 137 88 209)(29 136 89 208)(30 135 90 207)(31 174 91 216)(32 173 92 215)(33 172 93 214)(34 171 94 213)(35 170 95 212)(36 169 96 211)(37 132 97 218)(38 131 98 217)(39 130 99 222)(40 129 100 221)(41 128 101 220)(42 127 102 219)(43 75 198 147)(44 74 193 146)(45 73 194 145)(46 78 195 150)(47 77 196 149)(48 76 197 148)(49 64 188 161)(50 63 189 160)(51 62 190 159)(52 61 191 158)(53 66 192 157)(54 65 187 162)(55 68 184 140)(56 67 185 139)(57 72 186 144)(58 71 181 143)(59 70 182 142)(60 69 183 141)(103 224 115 236)(104 223 116 235)(105 228 117 240)(106 227 118 239)(107 226 119 238)(108 225 120 237)
(1 238 4 235)(2 237 5 240)(3 236 6 239)(7 33 10 36)(8 32 11 35)(9 31 12 34)(13 224 16 227)(14 223 17 226)(15 228 18 225)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 48 28 45)(26 47 29 44)(27 46 30 43)(37 50 40 53)(38 49 41 52)(39 54 42 51)(61 220 64 217)(62 219 65 222)(63 218 66 221)(67 199 70 202)(68 204 71 201)(69 203 72 200)(73 209 76 206)(74 208 77 205)(75 207 78 210)(79 182 82 185)(80 181 83 184)(81 186 84 183)(85 197 88 194)(86 196 89 193)(87 195 90 198)(91 231 94 234)(92 230 95 233)(93 229 96 232)(97 189 100 192)(98 188 101 191)(99 187 102 190)(103 179 106 176)(104 178 107 175)(105 177 108 180)(109 213 112 216)(110 212 113 215)(111 211 114 214)(115 167 118 164)(116 166 119 163)(117 165 120 168)(121 143 124 140)(122 142 125 139)(123 141 126 144)(127 162 130 159)(128 161 131 158)(129 160 132 157)(133 146 136 149)(134 145 137 148)(135 150 138 147)(151 171 154 174)(152 170 155 173)(153 169 156 172)
(1 25 95 81 97)(2 26 96 82 98)(3 27 91 83 99)(4 28 92 84 100)(5 29 93 79 101)(6 30 94 80 102)(7 56 49 225 196)(8 57 50 226 197)(9 58 51 227 198)(10 59 52 228 193)(11 60 53 223 194)(12 55 54 224 195)(13 87 31 23 39)(14 88 32 24 40)(15 89 33 19 41)(16 90 34 20 42)(17 85 35 21 37)(18 86 36 22 38)(43 234 181 190 239)(44 229 182 191 240)(45 230 183 192 235)(46 231 184 187 236)(47 232 185 188 237)(48 233 186 189 238)(61 117 146 114 70)(62 118 147 109 71)(63 119 148 110 72)(64 120 149 111 67)(65 115 150 112 68)(66 116 145 113 69)(73 155 141 157 104)(74 156 142 158 105)(75 151 143 159 106)(76 152 144 160 107)(77 153 139 161 108)(78 154 140 162 103)(121 130 179 210 174)(122 131 180 205 169)(123 132 175 206 170)(124 127 176 207 171)(125 128 177 208 172)(126 129 178 209 173)(133 211 199 217 168)(134 212 200 218 163)(135 213 201 219 164)(136 214 202 220 165)(137 215 203 221 166)(138 216 204 222 167)
(1 63 4 66)(2 64 5 61)(3 65 6 62)(7 172 10 169)(8 173 11 170)(9 174 12 171)(13 162 16 159)(14 157 17 160)(15 158 18 161)(19 74 22 77)(20 75 23 78)(21 76 24 73)(25 72 28 69)(26 67 29 70)(27 68 30 71)(31 154 34 151)(32 155 35 152)(33 156 36 153)(37 107 40 104)(38 108 41 105)(39 103 42 106)(43 204 46 201)(44 199 47 202)(45 200 48 203)(49 177 52 180)(50 178 53 175)(51 179 54 176)(55 207 58 210)(56 208 59 205)(57 209 60 206)(79 146 82 149)(80 147 83 150)(81 148 84 145)(85 144 88 141)(86 139 89 142)(87 140 90 143)(91 112 94 109)(92 113 95 110)(93 114 96 111)(97 119 100 116)(98 120 101 117)(99 115 102 118)(121 195 124 198)(122 196 125 193)(123 197 126 194)(127 227 130 224)(128 228 131 225)(129 223 132 226)(133 185 136 182)(134 186 137 183)(135 181 138 184)(163 189 166 192)(164 190 167 187)(165 191 168 188)(211 232 214 229)(212 233 215 230)(213 234 216 231)(217 237 220 240)(218 238 221 235)(219 239 222 236)
G:=sub<Sym(240)| (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), (1,163,17,175)(2,168,18,180)(3,167,13,179)(4,166,14,178)(5,165,15,177)(6,164,16,176)(7,111,232,153)(8,110,233,152)(9,109,234,151)(10,114,229,156)(11,113,230,155)(12,112,231,154)(19,125,79,202)(20,124,80,201)(21,123,81,200)(22,122,82,199)(23,121,83,204)(24,126,84,203)(25,134,85,206)(26,133,86,205)(27,138,87,210)(28,137,88,209)(29,136,89,208)(30,135,90,207)(31,174,91,216)(32,173,92,215)(33,172,93,214)(34,171,94,213)(35,170,95,212)(36,169,96,211)(37,132,97,218)(38,131,98,217)(39,130,99,222)(40,129,100,221)(41,128,101,220)(42,127,102,219)(43,75,198,147)(44,74,193,146)(45,73,194,145)(46,78,195,150)(47,77,196,149)(48,76,197,148)(49,64,188,161)(50,63,189,160)(51,62,190,159)(52,61,191,158)(53,66,192,157)(54,65,187,162)(55,68,184,140)(56,67,185,139)(57,72,186,144)(58,71,181,143)(59,70,182,142)(60,69,183,141)(103,224,115,236)(104,223,116,235)(105,228,117,240)(106,227,118,239)(107,226,119,238)(108,225,120,237), (1,238,4,235)(2,237,5,240)(3,236,6,239)(7,33,10,36)(8,32,11,35)(9,31,12,34)(13,224,16,227)(14,223,17,226)(15,228,18,225)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,48,28,45)(26,47,29,44)(27,46,30,43)(37,50,40,53)(38,49,41,52)(39,54,42,51)(61,220,64,217)(62,219,65,222)(63,218,66,221)(67,199,70,202)(68,204,71,201)(69,203,72,200)(73,209,76,206)(74,208,77,205)(75,207,78,210)(79,182,82,185)(80,181,83,184)(81,186,84,183)(85,197,88,194)(86,196,89,193)(87,195,90,198)(91,231,94,234)(92,230,95,233)(93,229,96,232)(97,189,100,192)(98,188,101,191)(99,187,102,190)(103,179,106,176)(104,178,107,175)(105,177,108,180)(109,213,112,216)(110,212,113,215)(111,211,114,214)(115,167,118,164)(116,166,119,163)(117,165,120,168)(121,143,124,140)(122,142,125,139)(123,141,126,144)(127,162,130,159)(128,161,131,158)(129,160,132,157)(133,146,136,149)(134,145,137,148)(135,150,138,147)(151,171,154,174)(152,170,155,173)(153,169,156,172), (1,25,95,81,97)(2,26,96,82,98)(3,27,91,83,99)(4,28,92,84,100)(5,29,93,79,101)(6,30,94,80,102)(7,56,49,225,196)(8,57,50,226,197)(9,58,51,227,198)(10,59,52,228,193)(11,60,53,223,194)(12,55,54,224,195)(13,87,31,23,39)(14,88,32,24,40)(15,89,33,19,41)(16,90,34,20,42)(17,85,35,21,37)(18,86,36,22,38)(43,234,181,190,239)(44,229,182,191,240)(45,230,183,192,235)(46,231,184,187,236)(47,232,185,188,237)(48,233,186,189,238)(61,117,146,114,70)(62,118,147,109,71)(63,119,148,110,72)(64,120,149,111,67)(65,115,150,112,68)(66,116,145,113,69)(73,155,141,157,104)(74,156,142,158,105)(75,151,143,159,106)(76,152,144,160,107)(77,153,139,161,108)(78,154,140,162,103)(121,130,179,210,174)(122,131,180,205,169)(123,132,175,206,170)(124,127,176,207,171)(125,128,177,208,172)(126,129,178,209,173)(133,211,199,217,168)(134,212,200,218,163)(135,213,201,219,164)(136,214,202,220,165)(137,215,203,221,166)(138,216,204,222,167), (1,63,4,66)(2,64,5,61)(3,65,6,62)(7,172,10,169)(8,173,11,170)(9,174,12,171)(13,162,16,159)(14,157,17,160)(15,158,18,161)(19,74,22,77)(20,75,23,78)(21,76,24,73)(25,72,28,69)(26,67,29,70)(27,68,30,71)(31,154,34,151)(32,155,35,152)(33,156,36,153)(37,107,40,104)(38,108,41,105)(39,103,42,106)(43,204,46,201)(44,199,47,202)(45,200,48,203)(49,177,52,180)(50,178,53,175)(51,179,54,176)(55,207,58,210)(56,208,59,205)(57,209,60,206)(79,146,82,149)(80,147,83,150)(81,148,84,145)(85,144,88,141)(86,139,89,142)(87,140,90,143)(91,112,94,109)(92,113,95,110)(93,114,96,111)(97,119,100,116)(98,120,101,117)(99,115,102,118)(121,195,124,198)(122,196,125,193)(123,197,126,194)(127,227,130,224)(128,228,131,225)(129,223,132,226)(133,185,136,182)(134,186,137,183)(135,181,138,184)(163,189,166,192)(164,190,167,187)(165,191,168,188)(211,232,214,229)(212,233,215,230)(213,234,216,231)(217,237,220,240)(218,238,221,235)(219,239,222,236)>;
G:=Group( (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), (1,163,17,175)(2,168,18,180)(3,167,13,179)(4,166,14,178)(5,165,15,177)(6,164,16,176)(7,111,232,153)(8,110,233,152)(9,109,234,151)(10,114,229,156)(11,113,230,155)(12,112,231,154)(19,125,79,202)(20,124,80,201)(21,123,81,200)(22,122,82,199)(23,121,83,204)(24,126,84,203)(25,134,85,206)(26,133,86,205)(27,138,87,210)(28,137,88,209)(29,136,89,208)(30,135,90,207)(31,174,91,216)(32,173,92,215)(33,172,93,214)(34,171,94,213)(35,170,95,212)(36,169,96,211)(37,132,97,218)(38,131,98,217)(39,130,99,222)(40,129,100,221)(41,128,101,220)(42,127,102,219)(43,75,198,147)(44,74,193,146)(45,73,194,145)(46,78,195,150)(47,77,196,149)(48,76,197,148)(49,64,188,161)(50,63,189,160)(51,62,190,159)(52,61,191,158)(53,66,192,157)(54,65,187,162)(55,68,184,140)(56,67,185,139)(57,72,186,144)(58,71,181,143)(59,70,182,142)(60,69,183,141)(103,224,115,236)(104,223,116,235)(105,228,117,240)(106,227,118,239)(107,226,119,238)(108,225,120,237), (1,238,4,235)(2,237,5,240)(3,236,6,239)(7,33,10,36)(8,32,11,35)(9,31,12,34)(13,224,16,227)(14,223,17,226)(15,228,18,225)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,48,28,45)(26,47,29,44)(27,46,30,43)(37,50,40,53)(38,49,41,52)(39,54,42,51)(61,220,64,217)(62,219,65,222)(63,218,66,221)(67,199,70,202)(68,204,71,201)(69,203,72,200)(73,209,76,206)(74,208,77,205)(75,207,78,210)(79,182,82,185)(80,181,83,184)(81,186,84,183)(85,197,88,194)(86,196,89,193)(87,195,90,198)(91,231,94,234)(92,230,95,233)(93,229,96,232)(97,189,100,192)(98,188,101,191)(99,187,102,190)(103,179,106,176)(104,178,107,175)(105,177,108,180)(109,213,112,216)(110,212,113,215)(111,211,114,214)(115,167,118,164)(116,166,119,163)(117,165,120,168)(121,143,124,140)(122,142,125,139)(123,141,126,144)(127,162,130,159)(128,161,131,158)(129,160,132,157)(133,146,136,149)(134,145,137,148)(135,150,138,147)(151,171,154,174)(152,170,155,173)(153,169,156,172), (1,25,95,81,97)(2,26,96,82,98)(3,27,91,83,99)(4,28,92,84,100)(5,29,93,79,101)(6,30,94,80,102)(7,56,49,225,196)(8,57,50,226,197)(9,58,51,227,198)(10,59,52,228,193)(11,60,53,223,194)(12,55,54,224,195)(13,87,31,23,39)(14,88,32,24,40)(15,89,33,19,41)(16,90,34,20,42)(17,85,35,21,37)(18,86,36,22,38)(43,234,181,190,239)(44,229,182,191,240)(45,230,183,192,235)(46,231,184,187,236)(47,232,185,188,237)(48,233,186,189,238)(61,117,146,114,70)(62,118,147,109,71)(63,119,148,110,72)(64,120,149,111,67)(65,115,150,112,68)(66,116,145,113,69)(73,155,141,157,104)(74,156,142,158,105)(75,151,143,159,106)(76,152,144,160,107)(77,153,139,161,108)(78,154,140,162,103)(121,130,179,210,174)(122,131,180,205,169)(123,132,175,206,170)(124,127,176,207,171)(125,128,177,208,172)(126,129,178,209,173)(133,211,199,217,168)(134,212,200,218,163)(135,213,201,219,164)(136,214,202,220,165)(137,215,203,221,166)(138,216,204,222,167), (1,63,4,66)(2,64,5,61)(3,65,6,62)(7,172,10,169)(8,173,11,170)(9,174,12,171)(13,162,16,159)(14,157,17,160)(15,158,18,161)(19,74,22,77)(20,75,23,78)(21,76,24,73)(25,72,28,69)(26,67,29,70)(27,68,30,71)(31,154,34,151)(32,155,35,152)(33,156,36,153)(37,107,40,104)(38,108,41,105)(39,103,42,106)(43,204,46,201)(44,199,47,202)(45,200,48,203)(49,177,52,180)(50,178,53,175)(51,179,54,176)(55,207,58,210)(56,208,59,205)(57,209,60,206)(79,146,82,149)(80,147,83,150)(81,148,84,145)(85,144,88,141)(86,139,89,142)(87,140,90,143)(91,112,94,109)(92,113,95,110)(93,114,96,111)(97,119,100,116)(98,120,101,117)(99,115,102,118)(121,195,124,198)(122,196,125,193)(123,197,126,194)(127,227,130,224)(128,228,131,225)(129,223,132,226)(133,185,136,182)(134,186,137,183)(135,181,138,184)(163,189,166,192)(164,190,167,187)(165,191,168,188)(211,232,214,229)(212,233,215,230)(213,234,216,231)(217,237,220,240)(218,238,221,235)(219,239,222,236) );
G=PermutationGroup([[(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)], [(1,163,17,175),(2,168,18,180),(3,167,13,179),(4,166,14,178),(5,165,15,177),(6,164,16,176),(7,111,232,153),(8,110,233,152),(9,109,234,151),(10,114,229,156),(11,113,230,155),(12,112,231,154),(19,125,79,202),(20,124,80,201),(21,123,81,200),(22,122,82,199),(23,121,83,204),(24,126,84,203),(25,134,85,206),(26,133,86,205),(27,138,87,210),(28,137,88,209),(29,136,89,208),(30,135,90,207),(31,174,91,216),(32,173,92,215),(33,172,93,214),(34,171,94,213),(35,170,95,212),(36,169,96,211),(37,132,97,218),(38,131,98,217),(39,130,99,222),(40,129,100,221),(41,128,101,220),(42,127,102,219),(43,75,198,147),(44,74,193,146),(45,73,194,145),(46,78,195,150),(47,77,196,149),(48,76,197,148),(49,64,188,161),(50,63,189,160),(51,62,190,159),(52,61,191,158),(53,66,192,157),(54,65,187,162),(55,68,184,140),(56,67,185,139),(57,72,186,144),(58,71,181,143),(59,70,182,142),(60,69,183,141),(103,224,115,236),(104,223,116,235),(105,228,117,240),(106,227,118,239),(107,226,119,238),(108,225,120,237)], [(1,238,4,235),(2,237,5,240),(3,236,6,239),(7,33,10,36),(8,32,11,35),(9,31,12,34),(13,224,16,227),(14,223,17,226),(15,228,18,225),(19,59,22,56),(20,58,23,55),(21,57,24,60),(25,48,28,45),(26,47,29,44),(27,46,30,43),(37,50,40,53),(38,49,41,52),(39,54,42,51),(61,220,64,217),(62,219,65,222),(63,218,66,221),(67,199,70,202),(68,204,71,201),(69,203,72,200),(73,209,76,206),(74,208,77,205),(75,207,78,210),(79,182,82,185),(80,181,83,184),(81,186,84,183),(85,197,88,194),(86,196,89,193),(87,195,90,198),(91,231,94,234),(92,230,95,233),(93,229,96,232),(97,189,100,192),(98,188,101,191),(99,187,102,190),(103,179,106,176),(104,178,107,175),(105,177,108,180),(109,213,112,216),(110,212,113,215),(111,211,114,214),(115,167,118,164),(116,166,119,163),(117,165,120,168),(121,143,124,140),(122,142,125,139),(123,141,126,144),(127,162,130,159),(128,161,131,158),(129,160,132,157),(133,146,136,149),(134,145,137,148),(135,150,138,147),(151,171,154,174),(152,170,155,173),(153,169,156,172)], [(1,25,95,81,97),(2,26,96,82,98),(3,27,91,83,99),(4,28,92,84,100),(5,29,93,79,101),(6,30,94,80,102),(7,56,49,225,196),(8,57,50,226,197),(9,58,51,227,198),(10,59,52,228,193),(11,60,53,223,194),(12,55,54,224,195),(13,87,31,23,39),(14,88,32,24,40),(15,89,33,19,41),(16,90,34,20,42),(17,85,35,21,37),(18,86,36,22,38),(43,234,181,190,239),(44,229,182,191,240),(45,230,183,192,235),(46,231,184,187,236),(47,232,185,188,237),(48,233,186,189,238),(61,117,146,114,70),(62,118,147,109,71),(63,119,148,110,72),(64,120,149,111,67),(65,115,150,112,68),(66,116,145,113,69),(73,155,141,157,104),(74,156,142,158,105),(75,151,143,159,106),(76,152,144,160,107),(77,153,139,161,108),(78,154,140,162,103),(121,130,179,210,174),(122,131,180,205,169),(123,132,175,206,170),(124,127,176,207,171),(125,128,177,208,172),(126,129,178,209,173),(133,211,199,217,168),(134,212,200,218,163),(135,213,201,219,164),(136,214,202,220,165),(137,215,203,221,166),(138,216,204,222,167)], [(1,63,4,66),(2,64,5,61),(3,65,6,62),(7,172,10,169),(8,173,11,170),(9,174,12,171),(13,162,16,159),(14,157,17,160),(15,158,18,161),(19,74,22,77),(20,75,23,78),(21,76,24,73),(25,72,28,69),(26,67,29,70),(27,68,30,71),(31,154,34,151),(32,155,35,152),(33,156,36,153),(37,107,40,104),(38,108,41,105),(39,103,42,106),(43,204,46,201),(44,199,47,202),(45,200,48,203),(49,177,52,180),(50,178,53,175),(51,179,54,176),(55,207,58,210),(56,208,59,205),(57,209,60,206),(79,146,82,149),(80,147,83,150),(81,148,84,145),(85,144,88,141),(86,139,89,142),(87,140,90,143),(91,112,94,109),(92,113,95,110),(93,114,96,111),(97,119,100,116),(98,120,101,117),(99,115,102,118),(121,195,124,198),(122,196,125,193),(123,197,126,194),(127,227,130,224),(128,228,131,225),(129,223,132,226),(133,185,136,182),(134,186,137,183),(135,181,138,184),(163,189,166,192),(164,190,167,187),(165,191,168,188),(211,232,214,229),(212,233,215,230),(213,234,216,231),(217,237,220,240),(218,238,221,235),(219,239,222,236)]])
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 | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 2 | 6 | 6 | 10 | 10 | 12 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
60 irreducible representations
| dim | 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 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | C4○D20 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | Dic5.D6 | C30.C23 | D5×C3⋊D4 |
| kernel | C6.(D4×D5) | Dic3×Dic5 | D10⋊Dic3 | C5×C6.D4 | C30.38D4 | C2×C15⋊Q8 | C6×C5⋊D4 | C2×C5⋊D4 | C3×Dic5 | C6.D4 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | Dic5 | C6 | C10 | C23 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C6.(D4×D5) ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 13 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 |
| 32 | 7 | 0 | 0 |
| 54 | 29 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 17 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 39 | 47 | 0 | 0 |
| 39 | 22 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,47,0,0,0,0,13],[50,0,0,0,0,50,0,0,0,0,0,1,0,0,60,0],[32,54,0,0,7,29,0,0,0,0,0,1,0,0,1,0],[17,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[39,39,0,0,47,22,0,0,0,0,1,0,0,0,0,1] >;
C6.(D4×D5) in GAP, Magma, Sage, TeX
C_6.(D_4\times D_5)
% in TeX
G:=Group("C6.(D4xD5)"); // GroupNames label
G:=SmallGroup(480,610);
// by ID
G=gap.SmallGroup(480,610);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,590,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=d^5=1,c^2=e^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=a^3*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^3*c,e*d*e^-1=d^-1>;
// generators/relations