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## G = D6⋊(C4×D5)  order 480 = 25·3·5

### 3rd semidirect product of D6 and C4×D5 acting via C4×D5/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D6⋊(C4×D5)
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — D6⋊(C4×D5)
 Lower central C15 — 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=ebe=a3b, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 860 in 188 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×8], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×4], C12 [×3], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×3], C22×S3, C22×C6, C5×S3 [×2], C3×D5 [×2], C30 [×3], C4×D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], Dic15, C60, C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×C3⋊D4, S3×Dic5 [×2], C15⋊D4 [×4], D5×C12 [×2], C6×Dic5, C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, Dic54D4, Dic3×Dic5, D10⋊Dic3, C5×D6⋊C4, C30.4Q8, C2×S3×Dic5, C2×C15⋊D4, D5×C2×C12, D6⋊(C4×D5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, C2×C4×D5, D4×D5, D42D5, C4×C3⋊D4, C2×S3×D5, Dic54D4, D125D5, C4×S3×D5, 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)
(1 177)(2 176)(3 175)(4 180)(5 179)(6 178)(7 108)(8 107)(9 106)(10 105)(11 104)(12 103)(13 112)(14 111)(15 110)(16 109)(17 114)(18 113)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 100)(26 99)(27 98)(28 97)(29 102)(30 101)(31 187)(32 192)(33 191)(34 190)(35 189)(36 188)(37 181)(38 186)(39 185)(40 184)(41 183)(42 182)(43 201)(44 200)(45 199)(46 204)(47 203)(48 202)(49 195)(50 194)(51 193)(52 198)(53 197)(54 196)(55 148)(56 147)(57 146)(58 145)(59 150)(60 149)(61 157)(62 162)(63 161)(64 160)(65 159)(66 158)(67 151)(68 156)(69 155)(70 154)(71 153)(72 152)(73 171)(74 170)(75 169)(76 174)(77 173)(78 172)(79 165)(80 164)(81 163)(82 168)(83 167)(84 166)(85 238)(86 237)(87 236)(88 235)(89 240)(90 239)(115 208)(116 207)(117 206)(118 205)(119 210)(120 209)(121 212)(122 211)(123 216)(124 215)(125 214)(126 213)(127 222)(128 221)(129 220)(130 219)(131 218)(132 217)(133 226)(134 225)(135 224)(136 223)(137 228)(138 227)(139 230)(140 229)(141 234)(142 233)(143 232)(144 231)
(1 116 56 86)(2 117 57 87)(3 118 58 88)(4 119 59 89)(5 120 60 90)(6 115 55 85)(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 38 53 44 36)(2 39 54 45 31)(3 40 49 46 32)(4 41 50 47 33)(5 42 51 48 34)(6 37 52 43 35)(7 21 239 29 14)(8 22 240 30 15)(9 23 235 25 16)(10 24 236 26 17)(11 19 237 27 18)(12 20 238 28 13)(55 67 82 73 65)(56 68 83 74 66)(57 69 84 75 61)(58 70 79 76 62)(59 71 80 77 63)(60 72 81 78 64)(85 97 112 103 95)(86 98 113 104 96)(87 99 114 105 91)(88 100 109 106 92)(89 101 110 107 93)(90 102 111 108 94)(115 122 137 144 130)(116 123 138 139 131)(117 124 133 140 132)(118 125 134 141 127)(119 126 135 142 128)(120 121 136 143 129)(145 154 165 174 162)(146 155 166 169 157)(147 156 167 170 158)(148 151 168 171 159)(149 152 163 172 160)(150 153 164 173 161)(175 184 195 204 192)(176 185 196 199 187)(177 186 197 200 188)(178 181 198 201 189)(179 182 193 202 190)(180 183 194 203 191)(205 214 225 234 222)(206 215 226 229 217)(207 216 227 230 218)(208 211 228 231 219)(209 212 223 232 220)(210 213 224 233 221)
(1 36)(2 31)(3 32)(4 33)(5 34)(6 35)(7 26)(8 27)(9 28)(10 29)(11 30)(12 25)(13 16)(14 17)(15 18)(19 240)(20 235)(21 236)(22 237)(23 238)(24 239)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(55 65)(56 66)(57 61)(58 62)(59 63)(60 64)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(85 95)(86 96)(87 91)(88 92)(89 93)(90 94)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(115 130)(116 131)(117 132)(118 127)(119 128)(120 129)(121 143)(122 144)(123 139)(124 140)(125 141)(126 142)(145 159)(146 160)(147 161)(148 162)(149 157)(150 158)(151 174)(152 169)(153 170)(154 171)(155 172)(156 173)(163 166)(164 167)(165 168)(175 189)(176 190)(177 191)(178 192)(179 187)(180 188)(181 204)(182 199)(183 200)(184 201)(185 202)(186 203)(193 196)(194 197)(195 198)(205 219)(206 220)(207 221)(208 222)(209 217)(210 218)(211 234)(212 229)(213 230)(214 231)(215 232)(216 233)(223 226)(224 227)(225 228)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 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 2 3 4 4 4 4 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 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 6 6 10 10 2 2 2 5 5 5 5 6 6 30 30 30 30 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

72 irreducible representations

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

Matrix representation of D6⋊(C4×D5) in GL5(𝔽61)

 1 0 0 0 0 0 14 41 0 0 0 0 48 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 14 41 0 0 0 25 47 0 0 0 0 0 1 0 0 0 0 0 1
,
 11 0 0 0 0 0 11 56 0 0 0 0 50 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 17 60 0 0 0 1 0
,
 60 0 0 0 0 0 1 55 0 0 0 0 60 0 0 0 0 0 17 60 0 0 0 44 44

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,14,0,0,0,0,41,48,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,14,25,0,0,0,41,47,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,11,0,0,0,0,56,50,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,17,1,0,0,0,60,0],[60,0,0,0,0,0,1,0,0,0,0,55,60,0,0,0,0,0,17,44,0,0,0,60,44] >;

D6⋊(C4×D5) in GAP, Magma, Sage, TeX

D_6\rtimes (C_4\times D_5)
% in TeX

G:=Group("D6:(C4xD5)");
// GroupNames label

G:=SmallGroup(480,516);
// by ID

G=gap.SmallGroup(480,516);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,219,58,1356,18822]);
// 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=e*b*e=a^3*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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