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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D6⋊C4⋊D5
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — D6⋊C4⋊D5
 Lower central C15 — C2×C30 — D6⋊C4⋊D5
 Upper central C1 — C22 — C2×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, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C22.D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, S3×C10, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.28D6, C15⋊D4, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, 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, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C22.D4, C22×D5, C4○D12, 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)(43 166)(44 165)(45 164)(46 163)(47 168)(48 167)(49 170)(50 169)(51 174)(52 173)(53 172)(54 171)(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)(73 196)(74 195)(75 194)(76 193)(77 198)(78 197)(79 200)(80 199)(81 204)(82 203)(83 202)(84 201)(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)(103 226)(104 225)(105 224)(106 223)(107 228)(108 227)(109 230)(110 229)(111 234)(112 233)(113 232)(114 231)(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)(43 133 73 103)(44 134 74 104)(45 135 75 105)(46 136 76 106)(47 137 77 107)(48 138 78 108)(49 139 79 109)(50 140 80 110)(51 141 81 111)(52 142 82 112)(53 143 83 113)(54 144 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 32 47)(2 40 52 33 48)(3 41 53 34 43)(4 42 54 35 44)(5 37 49 36 45)(6 38 50 31 46)(7 21 17 239 30)(8 22 18 240 25)(9 23 13 235 26)(10 24 14 236 27)(11 19 15 237 28)(12 20 16 238 29)(55 72 84 65 74)(56 67 79 66 75)(57 68 80 61 76)(58 69 81 62 77)(59 70 82 63 78)(60 71 83 64 73)(85 102 114 95 104)(86 97 109 96 105)(87 98 110 91 106)(88 99 111 92 107)(89 100 112 93 108)(90 101 113 94 103)(115 121 144 130 134)(116 122 139 131 135)(117 123 140 132 136)(118 124 141 127 137)(119 125 142 128 138)(120 126 143 129 133)(145 155 174 162 168)(146 156 169 157 163)(147 151 170 158 164)(148 152 171 159 165)(149 153 172 160 166)(150 154 173 161 167)(175 185 204 192 198)(176 186 199 187 193)(177 181 200 188 194)(178 182 201 189 195)(179 183 202 190 196)(180 184 203 191 197)(205 215 234 222 228)(206 216 229 217 223)(207 211 230 218 224)(208 212 231 219 225)(209 213 232 220 226)(210 214 233 221 227)
(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 232)(8 233)(9 234)(10 229)(11 230)(12 231)(13 205)(14 206)(15 207)(16 208)(17 209)(18 210)(19 211)(20 212)(21 213)(22 214)(23 215)(24 216)(25 221)(26 222)(27 217)(28 218)(29 219)(30 220)(31 38)(32 39)(33 40)(34 41)(35 42)(36 37)(55 74)(56 75)(57 76)(58 77)(59 78)(60 73)(61 68)(62 69)(63 70)(64 71)(65 72)(66 67)(85 104)(86 105)(87 106)(88 107)(89 108)(90 103)(91 98)(92 99)(93 100)(94 101)(95 102)(96 97)(115 134)(116 135)(117 136)(118 137)(119 138)(120 133)(121 130)(122 131)(123 132)(124 127)(125 128)(126 129)(145 198)(146 193)(147 194)(148 195)(149 196)(150 197)(151 188)(152 189)(153 190)(154 191)(155 192)(156 187)(157 186)(158 181)(159 182)(160 183)(161 184)(162 185)(163 176)(164 177)(165 178)(166 179)(167 180)(168 175)(169 199)(170 200)(171 201)(172 202)(173 203)(174 204)(223 236)(224 237)(225 238)(226 239)(227 240)(228 235)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 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 6 6 6 6 10 ··· 10 10 10 10 10 12 12 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 10 10 12 2 2 2 10 10 12 60 60 2 2 2 2 2 10 10 10 10 2 ··· 2 12 12 12 12 2 2 2 2 10 10 10 10 4 4 4 4 4 4 12 12 12 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 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 D6 C4○D4 D10 D10 D10 C3⋊D4 C4○D12 C4○D20 S3×D5 D4×D5 D4⋊2D5 C2×S3×D5 D6.D10 D12⋊5D5 D5×C3⋊D4 kernel D6⋊C4⋊D5 D10⋊Dic3 D6⋊Dic5 C6.Dic10 C5×D6⋊C4 C30.4Q8 C2×C15⋊D4 D5×C2×C12 C2×C4×D5 C6×D5 D6⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C22×S3 D10 C10 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 4 2 2 2 4 8 8 2 2 2 2 4 4 4

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

 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 20 0 0 0 0 0 47
,
 0 60 0 0 0 0 60 0 0 0 0 0 0 0 31 60 0 0 0 0 45 30 0 0 0 0 0 0 2 35 0 0 0 0 40 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 19 0 0 0 0 60 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 0 0 0 0 0 60 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,20,47],[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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