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G = D6⋊C4⋊D5order 480 = 25·3·5

2nd semidirect product of D6⋊C4 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C42D5, (C6×D5).59D4, C6.135(D4×D5), D6⋊Dic515C2, (C2×C20).201D6, C30.147(C2×D4), C30.82(C4○D4), C6.37(C4○D20), (C2×C12).268D10, C30.4Q827C2, C6.Dic1024C2, (C22×D5).89D6, C10.40(C4○D12), C6.28(D42D5), D10⋊Dic314C2, D10.27(C3⋊D4), C36(D10.12D4), (C2×C30).137C23, (C2×C60).391C22, (C2×Dic5).181D6, (C2×Dic3).42D10, (C22×S3).14D10, C52(C23.28D6), C2.15(D125D5), C1513(C22.D4), C2.26(D6.D10), (C10×Dic3).85C22, (C6×Dic5).208C22, (C2×Dic15).106C22, (C2×C4×D5)⋊12S3, (D5×C2×C12)⋊20C2, (C5×D6⋊C4)⋊27C2, C2.17(D5×C3⋊D4), (C2×C4).131(S3×D5), (C2×C15⋊D4).5C2, C10.37(C2×C3⋊D4), C22.189(C2×S3×D5), (S3×C2×C10).29C22, (D5×C2×C6).105C22, (C2×C6).149(C22×D5), (C2×C10).149(C22×S3), SmallGroup(480,523)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D6⋊C4⋊D5
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — D6⋊C4⋊D5
C15C2×C30 — D6⋊C4⋊D5
C1C22C2×C4

Generators and relations for D6⋊C4⋊D5
 G = < a,b,c,d,e | a6=b2=c4=d5=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, cbc-1=a3b, bd=db, ebe=bc2, cd=dc, ce=ec, ede=d-1 >

Subgroups: 764 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3, C6 [×3], C6 [×2], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], D5 [×2], C10 [×3], C10, Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C2×C12 [×3], C22×S3, C22×C6, C5×S3, C3×D5 [×2], C30 [×3], C22.D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4 [×2], D6⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5 [×2], C6×D5 [×2], S3×C10 [×3], C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.28D6, C15⋊D4 [×2], D5×C12 [×2], C6×Dic5, C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, D10.12D4, D10⋊Dic3, D6⋊Dic5, C6.Dic10, C5×D6⋊C4, C30.4Q8, C2×C15⋊D4, D5×C2×C12, D6⋊C4⋊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, C22.D4, C22×D5, C4○D12 [×2], C2×C3⋊D4, S3×D5, C4○D20, D4×D5, D42D5, C23.28D6, C2×S3×D5, D10.12D4, D6.D10, D125D5, D5×C3⋊D4, D6⋊C4⋊D5

Smallest permutation representation of D6⋊C4⋊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)
(1 145)(2 150)(3 149)(4 148)(5 147)(6 146)(7 143)(8 142)(9 141)(10 140)(11 139)(12 144)(13 137)(14 136)(15 135)(16 134)(17 133)(18 138)(19 131)(20 130)(21 129)(22 128)(23 127)(24 132)(25 125)(26 124)(27 123)(28 122)(29 121)(30 126)(31 157)(32 162)(33 161)(34 160)(35 159)(36 158)(37 151)(38 156)(39 155)(40 154)(41 153)(42 152)(43 171)(44 170)(45 169)(46 174)(47 173)(48 172)(49 165)(50 164)(51 163)(52 168)(53 167)(54 166)(55 178)(56 177)(57 176)(58 175)(59 180)(60 179)(61 187)(62 192)(63 191)(64 190)(65 189)(66 188)(67 181)(68 186)(69 185)(70 184)(71 183)(72 182)(73 201)(74 200)(75 199)(76 204)(77 203)(78 202)(79 195)(80 194)(81 193)(82 198)(83 197)(84 196)(85 208)(86 207)(87 206)(88 205)(89 210)(90 209)(91 217)(92 222)(93 221)(94 220)(95 219)(96 218)(97 211)(98 216)(99 215)(100 214)(101 213)(102 212)(103 231)(104 230)(105 229)(106 234)(107 233)(108 232)(109 225)(110 224)(111 223)(112 228)(113 227)(114 226)(115 238)(116 237)(117 236)(118 235)(119 240)(120 239)
(1 118 58 88)(2 119 59 89)(3 120 60 90)(4 115 55 85)(5 116 56 86)(6 117 57 87)(7 199 232 169)(8 200 233 170)(9 201 234 171)(10 202 229 172)(11 203 230 173)(12 204 231 174)(13 195 228 165)(14 196 223 166)(15 197 224 167)(16 198 225 168)(17 193 226 163)(18 194 227 164)(19 191 218 161)(20 192 219 162)(21 187 220 157)(22 188 221 158)(23 189 222 159)(24 190 217 160)(25 181 214 151)(26 182 215 152)(27 183 216 153)(28 184 211 154)(29 185 212 155)(30 186 213 156)(31 132 61 91)(32 127 62 92)(33 128 63 93)(34 129 64 94)(35 130 65 95)(36 131 66 96)(37 122 67 97)(38 123 68 98)(39 124 69 99)(40 125 70 100)(41 126 71 101)(42 121 72 102)(43 144 73 103)(44 139 74 104)(45 140 75 105)(46 141 76 106)(47 142 77 107)(48 143 78 108)(49 134 79 109)(50 135 80 110)(51 136 81 111)(52 137 82 112)(53 138 83 113)(54 133 84 114)(145 238 175 208)(146 239 176 209)(147 240 177 210)(148 235 178 205)(149 236 179 206)(150 237 180 207)
(1 39 51 48 31)(2 40 52 43 32)(3 41 53 44 33)(4 42 54 45 34)(5 37 49 46 35)(6 38 50 47 36)(7 24 235 26 14)(8 19 236 27 15)(9 20 237 28 16)(10 21 238 29 17)(11 22 239 30 18)(12 23 240 25 13)(55 72 84 75 64)(56 67 79 76 65)(57 68 80 77 66)(58 69 81 78 61)(59 70 82 73 62)(60 71 83 74 63)(85 102 114 105 94)(86 97 109 106 95)(87 98 110 107 96)(88 99 111 108 91)(89 100 112 103 92)(90 101 113 104 93)(115 121 133 140 129)(116 122 134 141 130)(117 123 135 142 131)(118 124 136 143 132)(119 125 137 144 127)(120 126 138 139 128)(145 155 163 172 157)(146 156 164 173 158)(147 151 165 174 159)(148 152 166 169 160)(149 153 167 170 161)(150 154 168 171 162)(175 185 193 202 187)(176 186 194 203 188)(177 181 195 204 189)(178 182 196 199 190)(179 183 197 200 191)(180 184 198 201 192)(205 215 223 232 217)(206 216 224 233 218)(207 211 225 234 219)(208 212 226 229 220)(209 213 227 230 221)(210 214 228 231 222)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 215)(8 216)(9 211)(10 212)(11 213)(12 214)(13 228)(14 223)(15 224)(16 225)(17 226)(18 227)(19 206)(20 207)(21 208)(22 209)(23 210)(24 205)(25 231)(26 232)(27 233)(28 234)(29 229)(30 230)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)(55 64)(56 65)(57 66)(58 61)(59 62)(60 63)(67 76)(68 77)(69 78)(70 73)(71 74)(72 75)(85 94)(86 95)(87 96)(88 91)(89 92)(90 93)(97 106)(98 107)(99 108)(100 103)(101 104)(102 105)(115 129)(116 130)(117 131)(118 132)(119 127)(120 128)(121 140)(122 141)(123 142)(124 143)(125 144)(126 139)(145 187)(146 188)(147 189)(148 190)(149 191)(150 192)(151 204)(152 199)(153 200)(154 201)(155 202)(156 203)(157 175)(158 176)(159 177)(160 178)(161 179)(162 180)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 182)(170 183)(171 184)(172 185)(173 186)(174 181)(217 235)(218 236)(219 237)(220 238)(221 239)(222 240)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J12A12B12C12D12E12F12G12H15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order12222223444444455666666610···101010101012121212121212121515202020202020202030···3060···60
size1111101012222101012606022222101010102···212121212222210101010444444121212124···44···4

66 irreducible representations

dim1111111122222222222224444444
type+++++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10D10C3⋊D4C4○D12C4○D20S3×D5D4×D5D42D5C2×S3×D5D6.D10D125D5D5×C3⋊D4
kernelD6⋊C4⋊D5D10⋊Dic3D6⋊Dic5C6.Dic10C5×D6⋊C4C30.4Q8C2×C15⋊D4D5×C2×C12C2×C4×D5C6×D5D6⋊C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C22×S3D10C10C6C2×C4C6C6C22C2C2C2
# reps1111111112211142224882222444

Matrix representation of D6⋊C4⋊D5 in GL6(𝔽61)

6000000
0600000
001000
000100
00001320
0000047
,
0600000
6000000
00316000
00453000
0000235
00004059
,
5000000
0110000
0050000
0005000
0000600
0000060
,
100000
010000
00601900
00601800
000010
000001
,
100000
0600000
0060000
0060100
000010
000001

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,20,47],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,31,45,0,0,0,0,60,30,0,0,0,0,0,0,2,40,0,0,0,0,35,59],[50,0,0,0,0,0,0,11,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,19,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D6⋊C4⋊D5 in GAP, Magma, Sage, TeX

D_6\rtimes C_4\rtimes D_5
% in TeX

G:=Group("D6:C4:D5");
// GroupNames label

G:=SmallGroup(480,523);
// by ID

G=gap.SmallGroup(480,523);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^2=c^4=d^5=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^3*b,b*d=d*b,e*b*e=b*c^2,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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