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G = D6.(C4×D5)  order 480 = 25·3·5

3rd non-split extension by D6 of C4×D5 acting via C4×D5/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D6.(C4×D5)
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×S3×Dic5 — 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=1, e2=a3, bab=a-1, ac=ca, ad=da, ae=ea, cbc-1=a3b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 620 in 152 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×8], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×9], C23, C10 [×3], C10 [×2], Dic3 [×4], C12 [×4], D6 [×2], D6 [×2], C2×C6, C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×2], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30 [×3], C42⋊C2, C2×Dic5 [×2], C2×Dic5 [×6], C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C4×Dic5, C4×Dic5, C10.D4 [×2], C23.D5, C5×C22⋊C4, C22×Dic5, C422S3, S3×Dic5 [×4], C6×Dic5 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C23.11D10, Dic3×Dic5, D6⋊Dic5, Dic155C4, C12×Dic5, C5×D6⋊C4, C30.4Q8, C2×S3×Dic5, D6.(C4×D5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C4×D5 [×2], C22×D5, S3×C2×C4, C4○D12 [×2], S3×D5, C2×C4×D5, D42D5 [×2], C422S3, C2×S3×D5, C23.11D10, D125D5, C4×S3×D5, Dic3.D10, 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 55)(2 60)(3 59)(4 58)(5 57)(6 56)(7 230)(8 229)(9 234)(10 233)(11 232)(12 231)(13 226)(14 225)(15 224)(16 223)(17 228)(18 227)(19 215)(20 214)(21 213)(22 212)(23 211)(24 216)(25 221)(26 220)(27 219)(28 218)(29 217)(30 222)(31 64)(32 63)(33 62)(34 61)(35 66)(36 65)(37 70)(38 69)(39 68)(40 67)(41 72)(42 71)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 80)(50 79)(51 84)(52 83)(53 82)(54 81)(85 119)(86 118)(87 117)(88 116)(89 115)(90 120)(91 121)(92 126)(93 125)(94 124)(95 123)(96 122)(97 127)(98 132)(99 131)(100 130)(101 129)(102 128)(103 135)(104 134)(105 133)(106 138)(107 137)(108 136)(109 143)(110 142)(111 141)(112 140)(113 139)(114 144)(145 176)(146 175)(147 180)(148 179)(149 178)(150 177)(151 184)(152 183)(153 182)(154 181)(155 186)(156 185)(157 190)(158 189)(159 188)(160 187)(161 192)(162 191)(163 198)(164 197)(165 196)(166 195)(167 194)(168 193)(169 200)(170 199)(171 204)(172 203)(173 202)(174 201)(205 239)(206 238)(207 237)(208 236)(209 235)(210 240)
(1 116 56 86)(2 117 57 87)(3 118 58 88)(4 119 59 89)(5 120 60 90)(6 115 55 85)(7 202 232 172)(8 203 233 173)(9 204 234 174)(10 199 229 169)(11 200 230 170)(12 201 231 171)(13 198 228 168)(14 193 223 163)(15 194 224 164)(16 195 225 165)(17 196 226 166)(18 197 227 167)(19 183 213 153)(20 184 214 154)(21 185 215 155)(22 186 216 156)(23 181 211 151)(24 182 212 152)(25 189 219 159)(26 190 220 160)(27 191 221 161)(28 192 222 162)(29 187 217 157)(30 188 218 158)(31 121 61 91)(32 122 62 92)(33 123 63 93)(34 124 64 94)(35 125 65 95)(36 126 66 96)(37 127 67 97)(38 128 68 98)(39 129 69 99)(40 130 70 100)(41 131 71 101)(42 132 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)(145 235 175 205)(146 236 176 206)(147 237 177 207)(148 238 178 208)(149 239 179 209)(150 240 180 210)
(1 43 53 39 36)(2 44 54 40 31)(3 45 49 41 32)(4 46 50 42 33)(5 47 51 37 34)(6 48 52 38 35)(7 13 235 21 30)(8 14 236 22 25)(9 15 237 23 26)(10 16 238 24 27)(11 17 239 19 28)(12 18 240 20 29)(55 78 82 68 65)(56 73 83 69 66)(57 74 84 70 61)(58 75 79 71 62)(59 76 80 72 63)(60 77 81 67 64)(85 108 112 98 95)(86 103 113 99 96)(87 104 114 100 91)(88 105 109 101 92)(89 106 110 102 93)(90 107 111 97 94)(115 138 142 128 125)(116 133 143 129 126)(117 134 144 130 121)(118 135 139 131 122)(119 136 140 132 123)(120 137 141 127 124)(145 155 158 172 168)(146 156 159 173 163)(147 151 160 174 164)(148 152 161 169 165)(149 153 162 170 166)(150 154 157 171 167)(175 185 188 202 198)(176 186 189 203 193)(177 181 190 204 194)(178 182 191 199 195)(179 183 192 200 196)(180 184 187 201 197)(205 215 218 232 228)(206 216 219 233 223)(207 211 220 234 224)(208 212 221 229 225)(209 213 222 230 226)(210 214 217 231 227)
(1 166 4 163)(2 167 5 164)(3 168 6 165)(7 138 10 135)(8 133 11 136)(9 134 12 137)(13 115 16 118)(14 116 17 119)(15 117 18 120)(19 132 22 129)(20 127 23 130)(21 128 24 131)(25 143 28 140)(26 144 29 141)(27 139 30 142)(31 150 34 147)(32 145 35 148)(33 146 36 149)(37 151 40 154)(38 152 41 155)(39 153 42 156)(43 170 46 173)(44 171 47 174)(45 172 48 169)(49 158 52 161)(50 159 53 162)(51 160 54 157)(55 195 58 198)(56 196 59 193)(57 197 60 194)(61 180 64 177)(62 175 65 178)(63 176 66 179)(67 181 70 184)(68 182 71 185)(69 183 72 186)(73 200 76 203)(74 201 77 204)(75 202 78 199)(79 188 82 191)(80 189 83 192)(81 190 84 187)(85 225 88 228)(86 226 89 223)(87 227 90 224)(91 210 94 207)(92 205 95 208)(93 206 96 209)(97 211 100 214)(98 212 101 215)(99 213 102 216)(103 230 106 233)(104 231 107 234)(105 232 108 229)(109 218 112 221)(110 219 113 222)(111 220 114 217)(121 240 124 237)(122 235 125 238)(123 236 126 239)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 6A 6B 6C 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 12E ··· 12L 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 10 10 10 10 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 2 2 2 5 5 5 5 6 6 10 10 30 30 30 30 2 2 2 2 2 2 ··· 2 12 12 12 12 2 2 2 2 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 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D5 D6 D6 C4○D4 D10 D10 D10 C4×S3 C4×D5 C4○D12 S3×D5 D4⋊2D5 C2×S3×D5 D12⋊5D5 C4×S3×D5 Dic3.D10 kernel D6.(C4×D5) Dic3×Dic5 D6⋊Dic5 Dic15⋊5C4 C12×Dic5 C5×D6⋊C4 C30.4Q8 C2×S3×Dic5 S3×Dic5 C4×Dic5 D6⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C22×S3 Dic5 D6 C10 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 4 2 2 2 4 8 8 2 4 2 4 4 4

Matrix representation of D6.(C4×D5) in GL4(𝔽61) generated by

 1 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 1 1 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 9 18 0 0 43 52 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 1 0 0 0 0 17 60 0 0 1 0
,
 11 0 0 0 0 11 0 0 0 0 59 32 0 0 59 2
G:=sub<GL(4,GF(61))| [1,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,60,0,0,0,0,60,0,0,0,0,60],[9,43,0,0,18,52,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,17,1,0,0,60,0],[11,0,0,0,0,11,0,0,0,0,59,59,0,0,32,2] >;

D6.(C4×D5) in GAP, Magma, Sage, TeX

D_6.(C_4\times D_5)
% in TeX

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

G:=SmallGroup(480,474);
// by ID

G=gap.SmallGroup(480,474);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,219,58,1356,18822]);
// Polycyclic

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

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