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## G = C2×C6×C5⋊D4order 480 = 25·3·5

### Direct product of C2×C6 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C6×C5⋊D4
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — D5×C22×C6 — C2×C6×C5⋊D4
 Lower central C5 — C10 — C2×C6×C5⋊D4
 Upper central C1 — C22×C6 — C23×C6

Generators and relations for C2×C6×C5⋊D4
G = < a,b,c,d,e | a2=b6=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1360 in 472 conjugacy classes, 210 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C12, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, C24, Dic5, D10, D10, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C3×D5, C30, C30, C30, C22×D4, C2×Dic5, C5⋊D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C22×C12, C6×D4, C23×C6, C23×C6, C3×Dic5, C6×D5, C6×D5, C2×C30, C2×C30, C22×Dic5, C2×C5⋊D4, C23×D5, C23×C10, D4×C2×C6, C6×Dic5, C3×C5⋊D4, D5×C2×C6, D5×C2×C6, C22×C30, C22×C30, C22×C30, C22×C5⋊D4, C2×C6×Dic5, C6×C5⋊D4, D5×C22×C6, C23×C30, C2×C6×C5⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C24, D10, C3×D4, C22×C6, C3×D5, C22×D4, C5⋊D4, C22×D5, C6×D4, C23×C6, C6×D5, C2×C5⋊D4, C23×D5, D4×C2×C6, C3×C5⋊D4, D5×C2×C6, C22×C5⋊D4, C6×C5⋊D4, D5×C22×C6, C2×C6×C5⋊D4

Smallest permutation representation of C2×C6×C5⋊D4
On 240 points
Generators in S240
(1 190)(2 191)(3 192)(4 187)(5 188)(6 189)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 155)(14 156)(15 151)(16 152)(17 153)(18 154)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(25 233)(26 234)(27 229)(28 230)(29 231)(30 232)(31 124)(32 125)(33 126)(34 121)(35 122)(36 123)(43 136)(44 137)(45 138)(46 133)(47 134)(48 135)(49 66)(50 61)(51 62)(52 63)(53 64)(54 65)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(67 165)(68 166)(69 167)(70 168)(71 163)(72 164)(79 217)(80 218)(81 219)(82 220)(83 221)(84 222)(85 101)(86 102)(87 97)(88 98)(89 99)(90 100)(91 237)(92 238)(93 239)(94 240)(95 235)(96 236)(109 202)(110 203)(111 204)(112 199)(113 200)(114 201)(115 208)(116 209)(117 210)(118 205)(119 206)(120 207)(127 160)(128 161)(129 162)(130 157)(131 158)(132 159)(139 172)(140 173)(141 174)(142 169)(143 170)(144 171)(145 178)(146 179)(147 180)(148 175)(149 176)(150 177)(181 214)(182 215)(183 216)(184 211)(185 212)(186 213)(193 226)(194 227)(195 228)(196 223)(197 224)(198 225)
(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 27 44 14 115)(2 28 45 15 116)(3 29 46 16 117)(4 30 47 17 118)(5 25 48 18 119)(6 26 43 13 120)(7 174 90 176 211)(8 169 85 177 212)(9 170 86 178 213)(10 171 87 179 214)(11 172 88 180 215)(12 173 89 175 216)(19 162 76 62 225)(20 157 77 63 226)(21 158 78 64 227)(22 159 73 65 228)(23 160 74 66 223)(24 161 75 61 224)(31 96 80 110 67)(32 91 81 111 68)(33 92 82 112 69)(34 93 83 113 70)(35 94 84 114 71)(36 95 79 109 72)(37 130 59 52 193)(38 131 60 53 194)(39 132 55 54 195)(40 127 56 49 196)(41 128 57 50 197)(42 129 58 51 198)(97 146 181 106 144)(98 147 182 107 139)(99 148 183 108 140)(100 149 184 103 141)(101 150 185 104 142)(102 145 186 105 143)(121 239 221 200 168)(122 240 222 201 163)(123 235 217 202 164)(124 236 218 203 165)(125 237 219 204 166)(126 238 220 199 167)(133 152 210 192 231)(134 153 205 187 232)(135 154 206 188 233)(136 155 207 189 234)(137 156 208 190 229)(138 151 209 191 230)
(1 181 79 52)(2 182 80 53)(3 183 81 54)(4 184 82 49)(5 185 83 50)(6 186 84 51)(7 238 223 205)(8 239 224 206)(9 240 225 207)(10 235 226 208)(11 236 227 209)(12 237 228 210)(13 143 35 42)(14 144 36 37)(15 139 31 38)(16 140 32 39)(17 141 33 40)(18 142 34 41)(19 155 170 122)(20 156 171 123)(21 151 172 124)(22 152 173 125)(23 153 174 126)(24 154 169 121)(25 150 113 57)(26 145 114 58)(27 146 109 59)(28 147 110 60)(29 148 111 55)(30 149 112 56)(43 102 71 129)(44 97 72 130)(45 98 67 131)(46 99 68 132)(47 100 69 127)(48 101 70 128)(61 188 212 221)(62 189 213 222)(63 190 214 217)(64 191 215 218)(65 192 216 219)(66 187 211 220)(73 231 175 204)(74 232 176 199)(75 233 177 200)(76 234 178 201)(77 229 179 202)(78 230 180 203)(85 168 161 135)(86 163 162 136)(87 164 157 137)(88 165 158 138)(89 166 159 133)(90 167 160 134)(91 195 117 108)(92 196 118 103)(93 197 119 104)(94 198 120 105)(95 193 115 106)(96 194 116 107)
(1 82)(2 83)(3 84)(4 79)(5 80)(6 81)(7 179)(8 180)(9 175)(10 176)(11 177)(12 178)(13 68)(14 69)(15 70)(16 71)(17 72)(18 67)(19 159)(20 160)(21 161)(22 162)(23 157)(24 158)(25 96)(26 91)(27 92)(28 93)(29 94)(30 95)(31 48)(32 43)(33 44)(34 45)(35 46)(36 47)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(49 52)(50 53)(51 54)(55 198)(56 193)(57 194)(58 195)(59 196)(60 197)(61 64)(62 65)(63 66)(73 225)(74 226)(75 227)(76 228)(77 223)(78 224)(85 172)(86 173)(87 174)(88 169)(89 170)(90 171)(97 141)(98 142)(99 143)(100 144)(101 139)(102 140)(103 146)(104 147)(105 148)(106 149)(107 150)(108 145)(109 118)(110 119)(111 120)(112 115)(113 116)(114 117)(121 138)(122 133)(123 134)(124 135)(125 136)(126 137)(151 168)(152 163)(153 164)(154 165)(155 166)(156 167)(181 184)(182 185)(183 186)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(199 208)(200 209)(201 210)(202 205)(203 206)(204 207)(211 214)(212 215)(213 216)(229 238)(230 239)(231 240)(232 235)(233 236)(234 237)

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

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

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

156 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6N 6O ··· 6V 6W ··· 6AD 10A ··· 10AD 12A ··· 12H 15A 15B 15C 15D 30A ··· 30BH order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 6 ··· 6 6 ··· 6 10 ··· 10 12 ··· 12 15 15 15 15 30 ··· 30 size 1 1 ··· 1 2 2 2 2 10 10 10 10 1 1 10 10 10 10 2 2 1 ··· 1 2 ··· 2 10 ··· 10 2 ··· 2 10 ··· 10 2 2 2 2 2 ··· 2

156 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D5 D10 C3×D4 C3×D5 C5⋊D4 C6×D5 C3×C5⋊D4 kernel C2×C6×C5⋊D4 C2×C6×Dic5 C6×C5⋊D4 D5×C22×C6 C23×C30 C22×C5⋊D4 C22×Dic5 C2×C5⋊D4 C23×D5 C23×C10 C2×C30 C23×C6 C22×C6 C2×C10 C24 C2×C6 C23 C22 # reps 1 1 12 1 1 2 2 24 2 2 4 2 14 8 4 16 28 32

Matrix representation of C2×C6×C5⋊D4 in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 48 0 0 0 0 60 0 0 0 0 13 0 0 0 0 13
,
 1 0 0 0 0 1 0 0 0 0 18 60 0 0 19 60
,
 1 0 0 0 0 60 0 0 0 0 53 47 0 0 9 8
,
 1 0 0 0 0 1 0 0 0 0 43 1 0 0 43 18
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[48,0,0,0,0,60,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,18,19,0,0,60,60],[1,0,0,0,0,60,0,0,0,0,53,9,0,0,47,8],[1,0,0,0,0,1,0,0,0,0,43,43,0,0,1,18] >;

C2×C6×C5⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_5\rtimes D_4
% in TeX

G:=Group("C2xC6xC5:D4");
// GroupNames label

G:=SmallGroup(480,1149);
// by ID

G=gap.SmallGroup(480,1149);
# by ID

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

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

׿
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𝔽