Copied to
clipboard

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.D45D5, C30.214(C2×D4), C23.16(S3×D5), C6.79(C4○D20), C1515(C4.4D4), C52(C23.12D6), (C3×Dic5).19D4, (C22×D5).27D6, (C22×C6).24D10, (C22×C10).39D6, (Dic3×Dic5)⋊32C2, C30.136(C4○D4), C6.78(D42D5), C30.38D417C2, D10⋊Dic326C2, (C2×C30).176C23, (C2×Dic5).127D6, C37(Dic5.5D4), C10.51(D42S3), (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

C1C2×C30 — C6.(D4×D5)
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — C6.(D4×D5)
C15C2×C30 — C6.(D4×D5)
C1C22C23

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 [×3], C2 [×2], C3, C4 [×6], C22, C22 [×6], C5, C6 [×3], C6 [×2], C2×C4 [×5], D4 [×2], Q8 [×2], C23, C23, D5, C10 [×3], C10, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×3], Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30 [×3], C30, C4.4D4, Dic10 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C4×Dic3, C6.D4, C6.D4 [×3], C2×Dic6, C6×D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C6×D5 [×3], C2×C30, C2×C30 [×3], C4×Dic5, D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C23.12D6, C15⋊Q8 [×2], C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3 [×2], C2×Dic15 [×2], D5×C2×C6, C22×C30, Dic5.5D4, Dic3×Dic5, D10⋊Dic3 [×2], C5×C6.D4, C30.38D4, C2×C15⋊Q8, C6×C5⋊D4, C6.(D4×D5)
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C4.4D4, C22×D5, D42S3 [×2], C2×C3⋊D4, S3×D5, C4○D20, D4×D5, D42D5, C23.12D6, C2×S3×D5, Dic5.5D4, Dic5.D6, C30.C23, D5×C3⋊D4, C6.(D4×D5)

Smallest permutation representation of 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 76 232 143)(8 75 233 142)(9 74 234 141)(10 73 229 140)(11 78 230 139)(12 77 231 144)(19 125 79 207)(20 124 80 206)(21 123 81 205)(22 122 82 210)(23 121 83 209)(24 126 84 208)(25 169 85 201)(26 174 86 200)(27 173 87 199)(28 172 88 204)(29 171 89 203)(30 170 90 202)(31 133 91 216)(32 138 92 215)(33 137 93 214)(34 136 94 213)(35 135 95 212)(36 134 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)(43 110 198 152)(44 109 193 151)(45 114 194 156)(46 113 195 155)(47 112 196 154)(48 111 197 153)(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 68 184 145)(56 67 185 150)(57 72 186 149)(58 71 181 148)(59 70 182 147)(60 69 183 146)(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 28 10 25)(8 27 11 30)(9 26 12 29)(13 224 16 227)(14 223 17 226)(15 228 18 225)(19 59 22 56)(20 58 23 55)(21 57 24 60)(31 47 34 44)(32 46 35 43)(33 45 36 48)(37 50 40 53)(38 49 41 52)(39 54 42 51)(61 220 64 217)(62 219 65 222)(63 218 66 221)(67 210 70 207)(68 209 71 206)(69 208 72 205)(73 204 76 201)(74 203 77 200)(75 202 78 199)(79 182 82 185)(80 181 83 184)(81 186 84 183)(85 232 88 229)(86 231 89 234)(87 230 90 233)(91 196 94 193)(92 195 95 198)(93 194 96 197)(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)(121 148 124 145)(122 147 125 150)(123 146 126 149)(127 162 130 159)(128 161 131 158)(129 160 132 157)(133 151 136 154)(134 156 137 153)(135 155 138 152)(139 173 142 170)(140 172 143 169)(141 171 144 174)
(1 20 95 86 97)(2 21 96 87 98)(3 22 91 88 99)(4 23 92 89 100)(5 24 93 90 101)(6 19 94 85 102)(7 51 227 182 44)(8 52 228 183 45)(9 53 223 184 46)(10 54 224 185 47)(11 49 225 186 48)(12 50 226 181 43)(13 82 31 28 39)(14 83 32 29 40)(15 84 33 30 41)(16 79 34 25 42)(17 80 35 26 37)(18 81 36 27 38)(55 195 234 192 235)(56 196 229 187 236)(57 197 230 188 237)(58 198 231 189 238)(59 193 232 190 239)(60 194 233 191 240)(61 117 146 114 75)(62 118 147 109 76)(63 119 148 110 77)(64 120 149 111 78)(65 115 150 112 73)(66 116 145 113 74)(67 154 140 162 103)(68 155 141 157 104)(69 156 142 158 105)(70 151 143 159 106)(71 152 144 160 107)(72 153 139 161 108)(121 215 203 221 166)(122 216 204 222 167)(123 211 199 217 168)(124 212 200 218 163)(125 213 201 219 164)(126 214 202 220 165)(127 176 207 136 169)(128 177 208 137 170)(129 178 209 138 171)(130 179 210 133 172)(131 180 205 134 173)(132 175 206 135 174)
(1 63 4 66)(2 64 5 61)(3 65 6 62)(7 210 10 207)(8 205 11 208)(9 206 12 209)(13 162 16 159)(14 157 17 160)(15 158 18 161)(19 76 22 73)(20 77 23 74)(21 78 24 75)(25 70 28 67)(26 71 29 68)(27 72 30 69)(31 154 34 151)(32 155 35 152)(33 156 36 153)(37 107 40 104)(38 108 41 105)(39 103 42 106)(43 138 46 135)(44 133 47 136)(45 134 48 137)(49 177 52 180)(50 178 53 175)(51 179 54 176)(55 200 58 203)(56 201 59 204)(57 202 60 199)(79 143 82 140)(80 144 83 141)(81 139 84 142)(85 147 88 150)(86 148 89 145)(87 149 90 146)(91 112 94 109)(92 113 95 110)(93 114 96 111)(97 119 100 116)(98 120 101 117)(99 115 102 118)(121 234 124 231)(122 229 125 232)(123 230 126 233)(127 227 130 224)(128 228 131 225)(129 223 132 226)(163 189 166 192)(164 190 167 187)(165 191 168 188)(169 182 172 185)(170 183 173 186)(171 184 174 181)(193 216 196 213)(194 211 197 214)(195 212 198 215)(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)(7,76,232,143)(8,75,233,142)(9,74,234,141)(10,73,229,140)(11,78,230,139)(12,77,231,144)(19,125,79,207)(20,124,80,206)(21,123,81,205)(22,122,82,210)(23,121,83,209)(24,126,84,208)(25,169,85,201)(26,174,86,200)(27,173,87,199)(28,172,88,204)(29,171,89,203)(30,170,90,202)(31,133,91,216)(32,138,92,215)(33,137,93,214)(34,136,94,213)(35,135,95,212)(36,134,96,211)(37,132,97,218)(38,131,98,217)(39,130,99,222)(40,129,100,221)(41,128,101,220)(42,127,102,219)(43,110,198,152)(44,109,193,151)(45,114,194,156)(46,113,195,155)(47,112,196,154)(48,111,197,153)(49,64,188,161)(50,63,189,160)(51,62,190,159)(52,61,191,158)(53,66,192,157)(54,65,187,162)(55,68,184,145)(56,67,185,150)(57,72,186,149)(58,71,181,148)(59,70,182,147)(60,69,183,146)(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)(7,28,10,25)(8,27,11,30)(9,26,12,29)(13,224,16,227)(14,223,17,226)(15,228,18,225)(19,59,22,56)(20,58,23,55)(21,57,24,60)(31,47,34,44)(32,46,35,43)(33,45,36,48)(37,50,40,53)(38,49,41,52)(39,54,42,51)(61,220,64,217)(62,219,65,222)(63,218,66,221)(67,210,70,207)(68,209,71,206)(69,208,72,205)(73,204,76,201)(74,203,77,200)(75,202,78,199)(79,182,82,185)(80,181,83,184)(81,186,84,183)(85,232,88,229)(86,231,89,234)(87,230,90,233)(91,196,94,193)(92,195,95,198)(93,194,96,197)(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)(121,148,124,145)(122,147,125,150)(123,146,126,149)(127,162,130,159)(128,161,131,158)(129,160,132,157)(133,151,136,154)(134,156,137,153)(135,155,138,152)(139,173,142,170)(140,172,143,169)(141,171,144,174), (1,20,95,86,97)(2,21,96,87,98)(3,22,91,88,99)(4,23,92,89,100)(5,24,93,90,101)(6,19,94,85,102)(7,51,227,182,44)(8,52,228,183,45)(9,53,223,184,46)(10,54,224,185,47)(11,49,225,186,48)(12,50,226,181,43)(13,82,31,28,39)(14,83,32,29,40)(15,84,33,30,41)(16,79,34,25,42)(17,80,35,26,37)(18,81,36,27,38)(55,195,234,192,235)(56,196,229,187,236)(57,197,230,188,237)(58,198,231,189,238)(59,193,232,190,239)(60,194,233,191,240)(61,117,146,114,75)(62,118,147,109,76)(63,119,148,110,77)(64,120,149,111,78)(65,115,150,112,73)(66,116,145,113,74)(67,154,140,162,103)(68,155,141,157,104)(69,156,142,158,105)(70,151,143,159,106)(71,152,144,160,107)(72,153,139,161,108)(121,215,203,221,166)(122,216,204,222,167)(123,211,199,217,168)(124,212,200,218,163)(125,213,201,219,164)(126,214,202,220,165)(127,176,207,136,169)(128,177,208,137,170)(129,178,209,138,171)(130,179,210,133,172)(131,180,205,134,173)(132,175,206,135,174), (1,63,4,66)(2,64,5,61)(3,65,6,62)(7,210,10,207)(8,205,11,208)(9,206,12,209)(13,162,16,159)(14,157,17,160)(15,158,18,161)(19,76,22,73)(20,77,23,74)(21,78,24,75)(25,70,28,67)(26,71,29,68)(27,72,30,69)(31,154,34,151)(32,155,35,152)(33,156,36,153)(37,107,40,104)(38,108,41,105)(39,103,42,106)(43,138,46,135)(44,133,47,136)(45,134,48,137)(49,177,52,180)(50,178,53,175)(51,179,54,176)(55,200,58,203)(56,201,59,204)(57,202,60,199)(79,143,82,140)(80,144,83,141)(81,139,84,142)(85,147,88,150)(86,148,89,145)(87,149,90,146)(91,112,94,109)(92,113,95,110)(93,114,96,111)(97,119,100,116)(98,120,101,117)(99,115,102,118)(121,234,124,231)(122,229,125,232)(123,230,126,233)(127,227,130,224)(128,228,131,225)(129,223,132,226)(163,189,166,192)(164,190,167,187)(165,191,168,188)(169,182,172,185)(170,183,173,186)(171,184,174,181)(193,216,196,213)(194,211,197,214)(195,212,198,215)(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)(7,76,232,143)(8,75,233,142)(9,74,234,141)(10,73,229,140)(11,78,230,139)(12,77,231,144)(19,125,79,207)(20,124,80,206)(21,123,81,205)(22,122,82,210)(23,121,83,209)(24,126,84,208)(25,169,85,201)(26,174,86,200)(27,173,87,199)(28,172,88,204)(29,171,89,203)(30,170,90,202)(31,133,91,216)(32,138,92,215)(33,137,93,214)(34,136,94,213)(35,135,95,212)(36,134,96,211)(37,132,97,218)(38,131,98,217)(39,130,99,222)(40,129,100,221)(41,128,101,220)(42,127,102,219)(43,110,198,152)(44,109,193,151)(45,114,194,156)(46,113,195,155)(47,112,196,154)(48,111,197,153)(49,64,188,161)(50,63,189,160)(51,62,190,159)(52,61,191,158)(53,66,192,157)(54,65,187,162)(55,68,184,145)(56,67,185,150)(57,72,186,149)(58,71,181,148)(59,70,182,147)(60,69,183,146)(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)(7,28,10,25)(8,27,11,30)(9,26,12,29)(13,224,16,227)(14,223,17,226)(15,228,18,225)(19,59,22,56)(20,58,23,55)(21,57,24,60)(31,47,34,44)(32,46,35,43)(33,45,36,48)(37,50,40,53)(38,49,41,52)(39,54,42,51)(61,220,64,217)(62,219,65,222)(63,218,66,221)(67,210,70,207)(68,209,71,206)(69,208,72,205)(73,204,76,201)(74,203,77,200)(75,202,78,199)(79,182,82,185)(80,181,83,184)(81,186,84,183)(85,232,88,229)(86,231,89,234)(87,230,90,233)(91,196,94,193)(92,195,95,198)(93,194,96,197)(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)(121,148,124,145)(122,147,125,150)(123,146,126,149)(127,162,130,159)(128,161,131,158)(129,160,132,157)(133,151,136,154)(134,156,137,153)(135,155,138,152)(139,173,142,170)(140,172,143,169)(141,171,144,174), (1,20,95,86,97)(2,21,96,87,98)(3,22,91,88,99)(4,23,92,89,100)(5,24,93,90,101)(6,19,94,85,102)(7,51,227,182,44)(8,52,228,183,45)(9,53,223,184,46)(10,54,224,185,47)(11,49,225,186,48)(12,50,226,181,43)(13,82,31,28,39)(14,83,32,29,40)(15,84,33,30,41)(16,79,34,25,42)(17,80,35,26,37)(18,81,36,27,38)(55,195,234,192,235)(56,196,229,187,236)(57,197,230,188,237)(58,198,231,189,238)(59,193,232,190,239)(60,194,233,191,240)(61,117,146,114,75)(62,118,147,109,76)(63,119,148,110,77)(64,120,149,111,78)(65,115,150,112,73)(66,116,145,113,74)(67,154,140,162,103)(68,155,141,157,104)(69,156,142,158,105)(70,151,143,159,106)(71,152,144,160,107)(72,153,139,161,108)(121,215,203,221,166)(122,216,204,222,167)(123,211,199,217,168)(124,212,200,218,163)(125,213,201,219,164)(126,214,202,220,165)(127,176,207,136,169)(128,177,208,137,170)(129,178,209,138,171)(130,179,210,133,172)(131,180,205,134,173)(132,175,206,135,174), (1,63,4,66)(2,64,5,61)(3,65,6,62)(7,210,10,207)(8,205,11,208)(9,206,12,209)(13,162,16,159)(14,157,17,160)(15,158,18,161)(19,76,22,73)(20,77,23,74)(21,78,24,75)(25,70,28,67)(26,71,29,68)(27,72,30,69)(31,154,34,151)(32,155,35,152)(33,156,36,153)(37,107,40,104)(38,108,41,105)(39,103,42,106)(43,138,46,135)(44,133,47,136)(45,134,48,137)(49,177,52,180)(50,178,53,175)(51,179,54,176)(55,200,58,203)(56,201,59,204)(57,202,60,199)(79,143,82,140)(80,144,83,141)(81,139,84,142)(85,147,88,150)(86,148,89,145)(87,149,90,146)(91,112,94,109)(92,113,95,110)(93,114,96,111)(97,119,100,116)(98,120,101,117)(99,115,102,118)(121,234,124,231)(122,229,125,232)(123,230,126,233)(127,227,130,224)(128,228,131,225)(129,223,132,226)(163,189,166,192)(164,190,167,187)(165,191,168,188)(169,182,172,185)(170,183,173,186)(171,184,174,181)(193,216,196,213)(194,211,197,214)(195,212,198,215)(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),(7,76,232,143),(8,75,233,142),(9,74,234,141),(10,73,229,140),(11,78,230,139),(12,77,231,144),(19,125,79,207),(20,124,80,206),(21,123,81,205),(22,122,82,210),(23,121,83,209),(24,126,84,208),(25,169,85,201),(26,174,86,200),(27,173,87,199),(28,172,88,204),(29,171,89,203),(30,170,90,202),(31,133,91,216),(32,138,92,215),(33,137,93,214),(34,136,94,213),(35,135,95,212),(36,134,96,211),(37,132,97,218),(38,131,98,217),(39,130,99,222),(40,129,100,221),(41,128,101,220),(42,127,102,219),(43,110,198,152),(44,109,193,151),(45,114,194,156),(46,113,195,155),(47,112,196,154),(48,111,197,153),(49,64,188,161),(50,63,189,160),(51,62,190,159),(52,61,191,158),(53,66,192,157),(54,65,187,162),(55,68,184,145),(56,67,185,150),(57,72,186,149),(58,71,181,148),(59,70,182,147),(60,69,183,146),(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),(7,28,10,25),(8,27,11,30),(9,26,12,29),(13,224,16,227),(14,223,17,226),(15,228,18,225),(19,59,22,56),(20,58,23,55),(21,57,24,60),(31,47,34,44),(32,46,35,43),(33,45,36,48),(37,50,40,53),(38,49,41,52),(39,54,42,51),(61,220,64,217),(62,219,65,222),(63,218,66,221),(67,210,70,207),(68,209,71,206),(69,208,72,205),(73,204,76,201),(74,203,77,200),(75,202,78,199),(79,182,82,185),(80,181,83,184),(81,186,84,183),(85,232,88,229),(86,231,89,234),(87,230,90,233),(91,196,94,193),(92,195,95,198),(93,194,96,197),(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),(121,148,124,145),(122,147,125,150),(123,146,126,149),(127,162,130,159),(128,161,131,158),(129,160,132,157),(133,151,136,154),(134,156,137,153),(135,155,138,152),(139,173,142,170),(140,172,143,169),(141,171,144,174)], [(1,20,95,86,97),(2,21,96,87,98),(3,22,91,88,99),(4,23,92,89,100),(5,24,93,90,101),(6,19,94,85,102),(7,51,227,182,44),(8,52,228,183,45),(9,53,223,184,46),(10,54,224,185,47),(11,49,225,186,48),(12,50,226,181,43),(13,82,31,28,39),(14,83,32,29,40),(15,84,33,30,41),(16,79,34,25,42),(17,80,35,26,37),(18,81,36,27,38),(55,195,234,192,235),(56,196,229,187,236),(57,197,230,188,237),(58,198,231,189,238),(59,193,232,190,239),(60,194,233,191,240),(61,117,146,114,75),(62,118,147,109,76),(63,119,148,110,77),(64,120,149,111,78),(65,115,150,112,73),(66,116,145,113,74),(67,154,140,162,103),(68,155,141,157,104),(69,156,142,158,105),(70,151,143,159,106),(71,152,144,160,107),(72,153,139,161,108),(121,215,203,221,166),(122,216,204,222,167),(123,211,199,217,168),(124,212,200,218,163),(125,213,201,219,164),(126,214,202,220,165),(127,176,207,136,169),(128,177,208,137,170),(129,178,209,138,171),(130,179,210,133,172),(131,180,205,134,173),(132,175,206,135,174)], [(1,63,4,66),(2,64,5,61),(3,65,6,62),(7,210,10,207),(8,205,11,208),(9,206,12,209),(13,162,16,159),(14,157,17,160),(15,158,18,161),(19,76,22,73),(20,77,23,74),(21,78,24,75),(25,70,28,67),(26,71,29,68),(27,72,30,69),(31,154,34,151),(32,155,35,152),(33,156,36,153),(37,107,40,104),(38,108,41,105),(39,103,42,106),(43,138,46,135),(44,133,47,136),(45,134,48,137),(49,177,52,180),(50,178,53,175),(51,179,54,176),(55,200,58,203),(56,201,59,204),(57,202,60,199),(79,143,82,140),(80,144,83,141),(81,139,84,142),(85,147,88,150),(86,148,89,145),(87,149,90,146),(91,112,94,109),(92,113,95,110),(93,114,96,111),(97,119,100,116),(98,120,101,117),(99,115,102,118),(121,234,124,231),(122,229,125,232),(123,230,126,233),(127,227,130,224),(128,228,131,225),(129,223,132,226),(163,189,166,192),(164,190,167,187),(165,191,168,188),(169,182,172,185),(170,183,173,186),(171,184,174,181),(193,216,196,213),(194,211,197,214),(195,212,198,215),(217,237,220,240),(218,238,221,235),(219,239,222,236)])

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J12A12B15A15B20A···20H30A···30N
order12222234444444455666666610···10101010101212151520···2030···30
size1111420266101012303060222224420202···2444420204412···124···4

60 irreducible representations

dim11111112222222222244444444
type+++++++++++++++-++-+-
imageC1C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10C3⋊D4C4○D20D42S3S3×D5D4×D5D42D5C2×S3×D5Dic5.D6C30.C23D5×C3⋊D4
kernelC6.(D4×D5)Dic3×Dic5D10⋊Dic3C5×C6.D4C30.38D4C2×C15⋊Q8C6×C5⋊D4C2×C5⋊D4C3×Dic5C6.D4C2×Dic5C22×D5C22×C10C30C2×Dic3C22×C6Dic5C6C10C23C6C6C22C2C2C2
# reps11211111221114424822222444

Matrix representation of C6.(D4×D5) in GL4(𝔽61) generated by

60000
06000
00470
00013
,
50000
05000
00060
0010
,
32700
542900
0001
0010
,
176000
1000
0010
0001
,
394700
392200
0010
0001
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

׿
×
𝔽