direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C22×C12, C60⋊13C23, C30.70C24, C5⋊2(C23×C12), C15⋊9(C23×C4), C30⋊9(C22×C4), C20⋊3(C22×C6), (C22×C60)⋊22C2, C10⋊2(C22×C12), (C22×C20)⋊14C6, (C2×C60)⋊51C22, C10.2(C23×C6), (C23×D5).7C6, C6.70(C23×D5), C23.39(C6×D5), Dic5⋊3(C22×C6), (C6×D5).74C23, (C2×C30).380C23, (C3×Dic5)⋊11C23, (C6×Dic5)⋊39C22, (C22×Dic5)⋊13C6, D10.15(C22×C6), (C22×C6).136D10, (C22×C30).165C22, (C2×C30)⋊36(C2×C4), (C2×C20)⋊14(C2×C6), C2.1(D5×C22×C6), (C2×C6×Dic5)⋊21C2, (C2×C10)⋊13(C2×C12), C22.29(D5×C2×C6), (D5×C22×C6).10C2, (C2×Dic5)⋊12(C2×C6), (D5×C2×C6).156C22, (C2×C10).63(C22×C6), (C22×C10).52(C2×C6), (C22×D5).45(C2×C6), (C2×C6).376(C22×D5), SmallGroup(480,1136)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C22×C12 |
Generators and relations for D5×C22×C12
G = < a,b,c,d,e | a2=b2=c12=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1200 in 472 conjugacy classes, 290 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C22×C6, C22×C6, C3×D5, C30, C30, C23×C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, C22×C12, C22×C12, C23×C6, C3×Dic5, C60, C6×D5, C2×C30, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C23×C12, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, C22×C30, D5×C22×C4, D5×C2×C12, C2×C6×Dic5, C22×C60, D5×C22×C6, D5×C22×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C22×C4, C24, D10, C2×C12, C22×C6, C3×D5, C23×C4, C4×D5, C22×D5, C22×C12, C23×C6, C6×D5, C2×C4×D5, C23×D5, C23×C12, D5×C12, D5×C2×C6, D5×C22×C4, D5×C2×C12, D5×C22×C6, D5×C22×C12
►On 240 points
Generators in S240
(1 168)(2 157)(3 158)(4 159)(5 160)(6 161)(7 162)(8 163)(9 164)(10 165)(11 166)(12 167)(13 221)(14 222)(15 223)(16 224)(17 225)(18 226)(19 227)(20 228)(21 217)(22 218)(23 219)(24 220)(25 188)(26 189)(27 190)(28 191)(29 192)(30 181)(31 182)(32 183)(33 184)(34 185)(35 186)(36 187)(37 72)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(73 202)(74 203)(75 204)(76 193)(77 194)(78 195)(79 196)(80 197)(81 198)(82 199)(83 200)(84 201)(85 140)(86 141)(87 142)(88 143)(89 144)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(109 153)(110 154)(111 155)(112 156)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(121 174)(122 175)(123 176)(124 177)(125 178)(126 179)(127 180)(128 169)(129 170)(130 171)(131 172)(132 173)(205 234)(206 235)(207 236)(208 237)(209 238)(210 239)(211 240)(212 229)(213 230)(214 231)(215 232)(216 233)
(1 70)(2 71)(3 72)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 32)(14 33)(15 34)(16 35)(17 36)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 157)(49 81)(50 82)(51 83)(52 84)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(85 117)(86 118)(87 119)(88 120)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 198)(98 199)(99 200)(100 201)(101 202)(102 203)(103 204)(104 193)(105 194)(106 195)(107 196)(108 197)(121 231)(122 232)(123 233)(124 234)(125 235)(126 236)(127 237)(128 238)(129 239)(130 240)(131 229)(132 230)(133 154)(134 155)(135 156)(136 145)(137 146)(138 147)(139 148)(140 149)(141 150)(142 151)(143 152)(144 153)(169 209)(170 210)(171 211)(172 212)(173 213)(174 214)(175 215)(176 216)(177 205)(178 206)(179 207)(180 208)(181 219)(182 220)(183 221)(184 222)(185 223)(186 224)(187 225)(188 226)(189 227)(190 228)(191 217)(192 218)
(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 16 137 178 84)(2 17 138 179 73)(3 18 139 180 74)(4 19 140 169 75)(5 20 141 170 76)(6 21 142 171 77)(7 22 143 172 78)(8 23 144 173 79)(9 24 133 174 80)(10 13 134 175 81)(11 14 135 176 82)(12 15 136 177 83)(25 148 208 54 72)(26 149 209 55 61)(27 150 210 56 62)(28 151 211 57 63)(29 152 212 58 64)(30 153 213 59 65)(31 154 214 60 66)(32 155 215 49 67)(33 156 216 50 68)(34 145 205 51 69)(35 146 206 52 70)(36 147 207 53 71)(37 188 116 237 102)(38 189 117 238 103)(39 190 118 239 104)(40 191 119 240 105)(41 192 120 229 106)(42 181 109 230 107)(43 182 110 231 108)(44 183 111 232 97)(45 184 112 233 98)(46 185 113 234 99)(47 186 114 235 100)(48 187 115 236 101)(85 128 204 159 227)(86 129 193 160 228)(87 130 194 161 217)(88 131 195 162 218)(89 132 196 163 219)(90 121 197 164 220)(91 122 198 165 221)(92 123 199 166 222)(93 124 200 167 223)(94 125 201 168 224)(95 126 202 157 225)(96 127 203 158 226)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 97)(11 98)(12 99)(13 232)(14 233)(15 234)(16 235)(17 236)(18 237)(19 238)(20 239)(21 240)(22 229)(23 230)(24 231)(25 127)(26 128)(27 129)(28 130)(29 131)(30 132)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 73)(49 165)(50 166)(51 167)(52 168)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 204)(62 193)(63 194)(64 195)(65 196)(66 197)(67 198)(68 199)(69 200)(70 201)(71 202)(72 203)(85 149)(86 150)(87 151)(88 152)(89 153)(90 154)(91 155)(92 156)(93 145)(94 146)(95 147)(96 148)(109 144)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(169 189)(170 190)(171 191)(172 192)(173 181)(174 182)(175 183)(176 184)(177 185)(178 186)(179 187)(180 188)(205 223)(206 224)(207 225)(208 226)(209 227)(210 228)(211 217)(212 218)(213 219)(214 220)(215 221)(216 222)
G:=sub<Sym(240)| (1,168)(2,157)(3,158)(4,159)(5,160)(6,161)(7,162)(8,163)(9,164)(10,165)(11,166)(12,167)(13,221)(14,222)(15,223)(16,224)(17,225)(18,226)(19,227)(20,228)(21,217)(22,218)(23,219)(24,220)(25,188)(26,189)(27,190)(28,191)(29,192)(30,181)(31,182)(32,183)(33,184)(34,185)(35,186)(36,187)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(73,202)(74,203)(75,204)(76,193)(77,194)(78,195)(79,196)(80,197)(81,198)(82,199)(83,200)(84,201)(85,140)(86,141)(87,142)(88,143)(89,144)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(109,153)(110,154)(111,155)(112,156)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,174)(122,175)(123,176)(124,177)(125,178)(126,179)(127,180)(128,169)(129,170)(130,171)(131,172)(132,173)(205,234)(206,235)(207,236)(208,237)(209,238)(210,239)(211,240)(212,229)(213,230)(214,231)(215,232)(216,233), (1,70)(2,71)(3,72)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,32)(14,33)(15,34)(16,35)(17,36)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,157)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(85,117)(86,118)(87,119)(88,120)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,198)(98,199)(99,200)(100,201)(101,202)(102,203)(103,204)(104,193)(105,194)(106,195)(107,196)(108,197)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,229)(132,230)(133,154)(134,155)(135,156)(136,145)(137,146)(138,147)(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)(177,205)(178,206)(179,207)(180,208)(181,219)(182,220)(183,221)(184,222)(185,223)(186,224)(187,225)(188,226)(189,227)(190,228)(191,217)(192,218), (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), (1,16,137,178,84)(2,17,138,179,73)(3,18,139,180,74)(4,19,140,169,75)(5,20,141,170,76)(6,21,142,171,77)(7,22,143,172,78)(8,23,144,173,79)(9,24,133,174,80)(10,13,134,175,81)(11,14,135,176,82)(12,15,136,177,83)(25,148,208,54,72)(26,149,209,55,61)(27,150,210,56,62)(28,151,211,57,63)(29,152,212,58,64)(30,153,213,59,65)(31,154,214,60,66)(32,155,215,49,67)(33,156,216,50,68)(34,145,205,51,69)(35,146,206,52,70)(36,147,207,53,71)(37,188,116,237,102)(38,189,117,238,103)(39,190,118,239,104)(40,191,119,240,105)(41,192,120,229,106)(42,181,109,230,107)(43,182,110,231,108)(44,183,111,232,97)(45,184,112,233,98)(46,185,113,234,99)(47,186,114,235,100)(48,187,115,236,101)(85,128,204,159,227)(86,129,193,160,228)(87,130,194,161,217)(88,131,195,162,218)(89,132,196,163,219)(90,121,197,164,220)(91,122,198,165,221)(92,123,199,166,222)(93,124,200,167,223)(94,125,201,168,224)(95,126,202,157,225)(96,127,203,158,226), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,97)(11,98)(12,99)(13,232)(14,233)(15,234)(16,235)(17,236)(18,237)(19,238)(20,239)(21,240)(22,229)(23,230)(24,231)(25,127)(26,128)(27,129)(28,130)(29,131)(30,132)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,73)(49,165)(50,166)(51,167)(52,168)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(61,204)(62,193)(63,194)(64,195)(65,196)(66,197)(67,198)(68,199)(69,200)(70,201)(71,202)(72,203)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,155)(92,156)(93,145)(94,146)(95,147)(96,148)(109,144)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(169,189)(170,190)(171,191)(172,192)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188)(205,223)(206,224)(207,225)(208,226)(209,227)(210,228)(211,217)(212,218)(213,219)(214,220)(215,221)(216,222)>;
G:=Group( (1,168)(2,157)(3,158)(4,159)(5,160)(6,161)(7,162)(8,163)(9,164)(10,165)(11,166)(12,167)(13,221)(14,222)(15,223)(16,224)(17,225)(18,226)(19,227)(20,228)(21,217)(22,218)(23,219)(24,220)(25,188)(26,189)(27,190)(28,191)(29,192)(30,181)(31,182)(32,183)(33,184)(34,185)(35,186)(36,187)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(73,202)(74,203)(75,204)(76,193)(77,194)(78,195)(79,196)(80,197)(81,198)(82,199)(83,200)(84,201)(85,140)(86,141)(87,142)(88,143)(89,144)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(109,153)(110,154)(111,155)(112,156)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,174)(122,175)(123,176)(124,177)(125,178)(126,179)(127,180)(128,169)(129,170)(130,171)(131,172)(132,173)(205,234)(206,235)(207,236)(208,237)(209,238)(210,239)(211,240)(212,229)(213,230)(214,231)(215,232)(216,233), (1,70)(2,71)(3,72)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,32)(14,33)(15,34)(16,35)(17,36)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,157)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(85,117)(86,118)(87,119)(88,120)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,198)(98,199)(99,200)(100,201)(101,202)(102,203)(103,204)(104,193)(105,194)(106,195)(107,196)(108,197)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,229)(132,230)(133,154)(134,155)(135,156)(136,145)(137,146)(138,147)(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)(177,205)(178,206)(179,207)(180,208)(181,219)(182,220)(183,221)(184,222)(185,223)(186,224)(187,225)(188,226)(189,227)(190,228)(191,217)(192,218), (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), (1,16,137,178,84)(2,17,138,179,73)(3,18,139,180,74)(4,19,140,169,75)(5,20,141,170,76)(6,21,142,171,77)(7,22,143,172,78)(8,23,144,173,79)(9,24,133,174,80)(10,13,134,175,81)(11,14,135,176,82)(12,15,136,177,83)(25,148,208,54,72)(26,149,209,55,61)(27,150,210,56,62)(28,151,211,57,63)(29,152,212,58,64)(30,153,213,59,65)(31,154,214,60,66)(32,155,215,49,67)(33,156,216,50,68)(34,145,205,51,69)(35,146,206,52,70)(36,147,207,53,71)(37,188,116,237,102)(38,189,117,238,103)(39,190,118,239,104)(40,191,119,240,105)(41,192,120,229,106)(42,181,109,230,107)(43,182,110,231,108)(44,183,111,232,97)(45,184,112,233,98)(46,185,113,234,99)(47,186,114,235,100)(48,187,115,236,101)(85,128,204,159,227)(86,129,193,160,228)(87,130,194,161,217)(88,131,195,162,218)(89,132,196,163,219)(90,121,197,164,220)(91,122,198,165,221)(92,123,199,166,222)(93,124,200,167,223)(94,125,201,168,224)(95,126,202,157,225)(96,127,203,158,226), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,97)(11,98)(12,99)(13,232)(14,233)(15,234)(16,235)(17,236)(18,237)(19,238)(20,239)(21,240)(22,229)(23,230)(24,231)(25,127)(26,128)(27,129)(28,130)(29,131)(30,132)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,73)(49,165)(50,166)(51,167)(52,168)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(61,204)(62,193)(63,194)(64,195)(65,196)(66,197)(67,198)(68,199)(69,200)(70,201)(71,202)(72,203)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,155)(92,156)(93,145)(94,146)(95,147)(96,148)(109,144)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(169,189)(170,190)(171,191)(172,192)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188)(205,223)(206,224)(207,225)(208,226)(209,227)(210,228)(211,217)(212,218)(213,219)(214,220)(215,221)(216,222) );
G=PermutationGroup([[(1,168),(2,157),(3,158),(4,159),(5,160),(6,161),(7,162),(8,163),(9,164),(10,165),(11,166),(12,167),(13,221),(14,222),(15,223),(16,224),(17,225),(18,226),(19,227),(20,228),(21,217),(22,218),(23,219),(24,220),(25,188),(26,189),(27,190),(28,191),(29,192),(30,181),(31,182),(32,183),(33,184),(34,185),(35,186),(36,187),(37,72),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(73,202),(74,203),(75,204),(76,193),(77,194),(78,195),(79,196),(80,197),(81,198),(82,199),(83,200),(84,201),(85,140),(86,141),(87,142),(88,143),(89,144),(90,133),(91,134),(92,135),(93,136),(94,137),(95,138),(96,139),(109,153),(110,154),(111,155),(112,156),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152),(121,174),(122,175),(123,176),(124,177),(125,178),(126,179),(127,180),(128,169),(129,170),(130,171),(131,172),(132,173),(205,234),(206,235),(207,236),(208,237),(209,238),(210,239),(211,240),(212,229),(213,230),(214,231),(215,232),(216,233)], [(1,70),(2,71),(3,72),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,32),(14,33),(15,34),(16,35),(17,36),(18,25),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31),(37,158),(38,159),(39,160),(40,161),(41,162),(42,163),(43,164),(44,165),(45,166),(46,167),(47,168),(48,157),(49,81),(50,82),(51,83),(52,84),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(85,117),(86,118),(87,119),(88,120),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,198),(98,199),(99,200),(100,201),(101,202),(102,203),(103,204),(104,193),(105,194),(106,195),(107,196),(108,197),(121,231),(122,232),(123,233),(124,234),(125,235),(126,236),(127,237),(128,238),(129,239),(130,240),(131,229),(132,230),(133,154),(134,155),(135,156),(136,145),(137,146),(138,147),(139,148),(140,149),(141,150),(142,151),(143,152),(144,153),(169,209),(170,210),(171,211),(172,212),(173,213),(174,214),(175,215),(176,216),(177,205),(178,206),(179,207),(180,208),(181,219),(182,220),(183,221),(184,222),(185,223),(186,224),(187,225),(188,226),(189,227),(190,228),(191,217),(192,218)], [(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)], [(1,16,137,178,84),(2,17,138,179,73),(3,18,139,180,74),(4,19,140,169,75),(5,20,141,170,76),(6,21,142,171,77),(7,22,143,172,78),(8,23,144,173,79),(9,24,133,174,80),(10,13,134,175,81),(11,14,135,176,82),(12,15,136,177,83),(25,148,208,54,72),(26,149,209,55,61),(27,150,210,56,62),(28,151,211,57,63),(29,152,212,58,64),(30,153,213,59,65),(31,154,214,60,66),(32,155,215,49,67),(33,156,216,50,68),(34,145,205,51,69),(35,146,206,52,70),(36,147,207,53,71),(37,188,116,237,102),(38,189,117,238,103),(39,190,118,239,104),(40,191,119,240,105),(41,192,120,229,106),(42,181,109,230,107),(43,182,110,231,108),(44,183,111,232,97),(45,184,112,233,98),(46,185,113,234,99),(47,186,114,235,100),(48,187,115,236,101),(85,128,204,159,227),(86,129,193,160,228),(87,130,194,161,217),(88,131,195,162,218),(89,132,196,163,219),(90,121,197,164,220),(91,122,198,165,221),(92,123,199,166,222),(93,124,200,167,223),(94,125,201,168,224),(95,126,202,157,225),(96,127,203,158,226)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,97),(11,98),(12,99),(13,232),(14,233),(15,234),(16,235),(17,236),(18,237),(19,238),(20,239),(21,240),(22,229),(23,230),(24,231),(25,127),(26,128),(27,129),(28,130),(29,131),(30,132),(31,121),(32,122),(33,123),(34,124),(35,125),(36,126),(37,74),(38,75),(39,76),(40,77),(41,78),(42,79),(43,80),(44,81),(45,82),(46,83),(47,84),(48,73),(49,165),(50,166),(51,167),(52,168),(53,157),(54,158),(55,159),(56,160),(57,161),(58,162),(59,163),(60,164),(61,204),(62,193),(63,194),(64,195),(65,196),(66,197),(67,198),(68,199),(69,200),(70,201),(71,202),(72,203),(85,149),(86,150),(87,151),(88,152),(89,153),(90,154),(91,155),(92,156),(93,145),(94,146),(95,147),(96,148),(109,144),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(118,141),(119,142),(120,143),(169,189),(170,190),(171,191),(172,192),(173,181),(174,182),(175,183),(176,184),(177,185),(178,186),(179,187),(180,188),(205,223),(206,224),(207,225),(208,226),(209,227),(210,228),(211,217),(212,218),(213,219),(214,220),(215,221),(216,222)]])
192 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 6A | ··· | 6N | 6O | ··· | 6AD | 10A | ··· | 10N | 12A | ··· | 12P | 12Q | ··· | 12AF | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 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 | 5 | ··· | 5 | 1 | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
192 irreducible representations
| dim | 1 | 1 | 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 | C4 | C6 | C6 | C6 | C6 | C12 | D5 | D10 | D10 | C3×D5 | C4×D5 | C6×D5 | C6×D5 | D5×C12 |
| kernel | D5×C22×C12 | D5×C2×C12 | C2×C6×Dic5 | C22×C60 | D5×C22×C6 | D5×C22×C4 | D5×C2×C6 | C2×C4×D5 | C22×Dic5 | C22×C20 | C23×D5 | C22×D5 | C22×C12 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 2 | 16 | 24 | 2 | 2 | 2 | 32 | 2 | 12 | 2 | 4 | 16 | 24 | 4 | 32 |
Matrix representation of D5×C22×C12 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 21 | 0 |
| 0 | 0 | 0 | 21 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 43 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,60,0,0,0,0,21,0,0,0,0,21],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,43],[60,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
D5×C22×C12 in GAP, Magma, Sage, TeX
D_5\times C_2^2\times C_{12} % in TeX
G:=Group("D5xC2^2xC12"); // GroupNames label
G:=SmallGroup(480,1136);
// by ID
G=gap.SmallGroup(480,1136);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,192,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^12=d^5=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,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations