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## G = C6×Q8⋊2D5order 480 = 25·3·5

### Direct product of C6 and Q8⋊2D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×Q8⋊2D5
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — D5×C2×C12 — C6×Q8⋊2D5
 Lower central C5 — C10 — C6×Q8⋊2D5
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C6×Q82D5
G = < a,b,c,d,e | a6=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 976 in 328 conjugacy classes, 178 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C2×C4○D4, C4×D5, D20, C2×Dic5, C2×C20, C5×Q8, C22×D5, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C4×D5, C2×D20, Q82D5, Q8×C10, C6×C4○D4, D5×C12, C3×D20, C6×Dic5, C2×C60, Q8×C15, D5×C2×C6, C2×Q82D5, D5×C2×C12, C6×D20, C3×Q82D5, Q8×C30, C6×Q82D5
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C4○D4, C24, D10, C22×C6, C3×D5, C2×C4○D4, C22×D5, C3×C4○D4, C23×C6, C6×D5, Q82D5, C23×D5, C6×C4○D4, D5×C2×C6, C2×Q82D5, C3×Q82D5, D5×C22×C6, C6×Q82D5

Smallest permutation representation of C6×Q82D5
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)
(1 107 187 165)(2 108 188 166)(3 103 189 167)(4 104 190 168)(5 105 191 163)(6 106 192 164)(7 30 35 232)(8 25 36 233)(9 26 31 234)(10 27 32 229)(11 28 33 230)(12 29 34 231)(13 111 205 153)(14 112 206 154)(15 113 207 155)(16 114 208 156)(17 109 209 151)(18 110 210 152)(19 176 82 118)(20 177 83 119)(21 178 84 120)(22 179 79 115)(23 180 80 116)(24 175 81 117)(37 215 201 61)(38 216 202 62)(39 211 203 63)(40 212 204 64)(41 213 199 65)(42 214 200 66)(43 91 67 159)(44 92 68 160)(45 93 69 161)(46 94 70 162)(47 95 71 157)(48 96 72 158)(49 149 225 85)(50 150 226 86)(51 145 227 87)(52 146 228 88)(53 147 223 89)(54 148 224 90)(55 195 217 144)(56 196 218 139)(57 197 219 140)(58 198 220 141)(59 193 221 142)(60 194 222 143)(73 138 97 124)(74 133 98 125)(75 134 99 126)(76 135 100 121)(77 136 101 122)(78 137 102 123)(127 170 184 236)(128 171 185 237)(129 172 186 238)(130 173 181 239)(131 174 182 240)(132 169 183 235)
(1 217 187 55)(2 218 188 56)(3 219 189 57)(4 220 190 58)(5 221 191 59)(6 222 192 60)(7 216 35 62)(8 211 36 63)(9 212 31 64)(10 213 32 65)(11 214 33 66)(12 215 34 61)(13 235 205 169)(14 236 206 170)(15 237 207 171)(16 238 208 172)(17 239 209 173)(18 240 210 174)(19 228 82 52)(20 223 83 53)(21 224 84 54)(22 225 79 49)(23 226 80 50)(24 227 81 51)(25 39 233 203)(26 40 234 204)(27 41 229 199)(28 42 230 200)(29 37 231 201)(30 38 232 202)(43 97 67 73)(44 98 68 74)(45 99 69 75)(46 100 70 76)(47 101 71 77)(48 102 72 78)(85 179 149 115)(86 180 150 116)(87 175 145 117)(88 176 146 118)(89 177 147 119)(90 178 148 120)(91 138 159 124)(92 133 160 125)(93 134 161 126)(94 135 162 121)(95 136 157 122)(96 137 158 123)(103 197 167 140)(104 198 168 141)(105 193 163 142)(106 194 164 143)(107 195 165 144)(108 196 166 139)(109 181 151 130)(110 182 152 131)(111 183 153 132)(112 184 154 127)(113 185 155 128)(114 186 156 129)
(1 199 147 71 13)(2 200 148 72 14)(3 201 149 67 15)(4 202 150 68 16)(5 203 145 69 17)(6 204 146 70 18)(7 80 125 186 198)(8 81 126 181 193)(9 82 121 182 194)(10 83 122 183 195)(11 84 123 184 196)(12 79 124 185 197)(19 135 131 143 31)(20 136 132 144 32)(21 137 127 139 33)(22 138 128 140 34)(23 133 129 141 35)(24 134 130 142 36)(25 117 75 239 221)(26 118 76 240 222)(27 119 77 235 217)(28 120 78 236 218)(29 115 73 237 219)(30 116 74 238 220)(37 85 43 207 189)(38 86 44 208 190)(39 87 45 209 191)(40 88 46 210 192)(41 89 47 205 187)(42 90 48 206 188)(49 91 155 167 215)(50 92 156 168 216)(51 93 151 163 211)(52 94 152 164 212)(53 95 153 165 213)(54 96 154 166 214)(55 229 177 101 169)(56 230 178 102 170)(57 231 179 97 171)(58 232 180 98 172)(59 233 175 99 173)(60 234 176 100 174)(61 225 159 113 103)(62 226 160 114 104)(63 227 161 109 105)(64 228 162 110 106)(65 223 157 111 107)(66 224 158 112 108)
(1 183)(2 184)(3 185)(4 186)(5 181)(6 182)(7 68)(8 69)(9 70)(10 71)(11 72)(12 67)(13 195)(14 196)(15 197)(16 198)(17 193)(18 194)(19 88)(20 89)(21 90)(22 85)(23 86)(24 87)(25 93)(26 94)(27 95)(28 96)(29 91)(30 92)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 138)(38 133)(39 134)(40 135)(41 136)(42 137)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 111)(56 112)(57 113)(58 114)(59 109)(60 110)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(73 215)(74 216)(75 211)(76 212)(77 213)(78 214)(79 149)(80 150)(81 145)(82 146)(83 147)(84 148)(103 171)(104 172)(105 173)(106 174)(107 169)(108 170)(121 204)(122 199)(123 200)(124 201)(125 202)(126 203)(127 188)(128 189)(129 190)(130 191)(131 192)(132 187)(139 206)(140 207)(141 208)(142 209)(143 210)(144 205)(151 221)(152 222)(153 217)(154 218)(155 219)(156 220)(157 229)(158 230)(159 231)(160 232)(161 233)(162 234)(163 239)(164 240)(165 235)(166 236)(167 237)(168 238)(175 227)(176 228)(177 223)(178 224)(179 225)(180 226)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A ··· 4F 4G 4H 4I 4J 5A 5B 6A ··· 6F 6G ··· 6R 10A ··· 10F 12A ··· 12L 12M ··· 12T 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 ··· 2 3 3 4 ··· 4 4 4 4 4 5 5 6 ··· 6 6 ··· 6 10 ··· 10 12 ··· 12 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 ··· 10 1 1 2 ··· 2 5 5 5 5 2 2 1 ··· 1 10 ··· 10 2 ··· 2 2 ··· 2 5 ··· 5 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D5 C4○D4 D10 D10 C3×D5 C3×C4○D4 C6×D5 C6×D5 Q8⋊2D5 C3×Q8⋊2D5 kernel C6×Q8⋊2D5 D5×C2×C12 C6×D20 C3×Q8⋊2D5 Q8×C30 C2×Q8⋊2D5 C2×C4×D5 C2×D20 Q8⋊2D5 Q8×C10 C6×Q8 C30 C2×C12 C3×Q8 C2×Q8 C10 C2×C4 Q8 C6 C2 # reps 1 3 3 8 1 2 6 6 16 2 2 4 6 8 4 8 12 16 4 8

Matrix representation of C6×Q82D5 in GL4(𝔽61) generated by

 48 0 0 0 0 48 0 0 0 0 13 0 0 0 0 13
,
 60 0 0 0 0 60 0 0 0 0 11 0 0 0 0 50
,
 60 0 0 0 0 60 0 0 0 0 0 1 0 0 60 0
,
 17 60 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 17 60 0 0 44 44 0 0 0 0 0 11 0 0 50 0
G:=sub<GL(4,GF(61))| [48,0,0,0,0,48,0,0,0,0,13,0,0,0,0,13],[60,0,0,0,0,60,0,0,0,0,11,0,0,0,0,50],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[17,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[17,44,0,0,60,44,0,0,0,0,0,50,0,0,11,0] >;

C6×Q82D5 in GAP, Magma, Sage, TeX

C_6\times Q_8\rtimes_2D_5
% in TeX

G:=Group("C6xQ8:2D5");
// GroupNames label

G:=SmallGroup(480,1143);
// by ID

G=gap.SmallGroup(480,1143);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,1571,409,192,18822]);
// Polycyclic

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

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