direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C6×C5⋊D4, C30.83C24, C10⋊3(C6×D4), C30⋊17(C2×D4), (C2×C30)⋊33D4, C24⋊8(C3×D5), (C23×C6)⋊4D5, C23⋊5(C6×D5), (C23×C30)⋊7C2, (C23×D5)⋊8C6, C15⋊18(C22×D4), (C23×C10)⋊12C6, (C2×C30)⋊12C23, D10⋊3(C22×C6), (C6×D5)⋊12C23, (C22×C6)⋊13D10, C6.83(C23×D5), C10.15(C23×C6), Dic5⋊2(C22×C6), (C22×C30)⋊21C22, (C22×Dic5)⋊12C6, (C3×Dic5)⋊10C23, (C6×Dic5)⋊38C22, C5⋊3(D4×C2×C6), C22⋊3(D5×C2×C6), (C2×C10)⋊12(C3×D4), (C2×C6×Dic5)⋊20C2, (D5×C2×C6)⋊23C22, (D5×C22×C6)⋊11C2, (C2×C6)⋊8(C22×D5), C2.15(D5×C22×C6), (C22×C10)⋊9(C2×C6), (C2×C10)⋊5(C22×C6), (C22×D5)⋊8(C2×C6), (C2×Dic5)⋊11(C2×C6), SmallGroup(480,1149)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1360 in 472 conjugacy classes, 210 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×11], C22 [×28], C5, C6, C6 [×6], C6 [×8], C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], D5 [×4], C10, C10 [×6], C10 [×4], C12 [×4], C2×C6 [×11], C2×C6 [×28], C15, C22×C4, C2×D4 [×12], C24, C24, Dic5 [×4], D10 [×4], D10 [×12], C2×C10 [×11], C2×C10 [×12], C2×C12 [×6], C3×D4 [×16], C22×C6, C22×C6 [×6], C22×C6 [×14], C3×D5 [×4], C30, C30 [×6], C30 [×4], C22×D4, C2×Dic5 [×6], C5⋊D4 [×16], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C22×C12, C6×D4 [×12], C23×C6, C23×C6, C3×Dic5 [×4], C6×D5 [×4], C6×D5 [×12], C2×C30 [×11], C2×C30 [×12], C22×Dic5, C2×C5⋊D4 [×12], C23×D5, C23×C10, D4×C2×C6, C6×Dic5 [×6], C3×C5⋊D4 [×16], D5×C2×C6 [×6], D5×C2×C6 [×4], C22×C30, C22×C30 [×6], C22×C30 [×4], C22×C5⋊D4, C2×C6×Dic5, C6×C5⋊D4 [×12], D5×C22×C6, C23×C30, C2×C6×C5⋊D4
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], D5, C2×C6 [×35], C2×D4 [×6], C24, D10 [×7], C3×D4 [×4], C22×C6 [×15], C3×D5, C22×D4, C5⋊D4 [×4], C22×D5 [×7], C6×D4 [×6], C23×C6, C6×D5 [×7], C2×C5⋊D4 [×6], C23×D5, D4×C2×C6, C3×C5⋊D4 [×4], D5×C2×C6 [×7], C22×C5⋊D4, C6×C5⋊D4 [×6], D5×C22×C6, C2×C6×C5⋊D4
Generators and relations
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 >
►On 240 points
Generators in S240
(1 190)(2 191)(3 192)(4 187)(5 188)(6 189)(7 105)(8 106)(9 107)(10 108)(11 103)(12 104)(13 231)(14 232)(15 233)(16 234)(17 229)(18 230)(19 41)(20 42)(21 37)(22 38)(23 39)(24 40)(25 237)(26 238)(27 239)(28 240)(29 235)(30 236)(31 124)(32 125)(33 126)(34 121)(35 122)(36 123)(43 77)(44 78)(45 73)(46 74)(47 75)(48 76)(49 155)(50 156)(51 151)(52 152)(53 153)(54 154)(55 148)(56 149)(57 150)(58 145)(59 146)(60 147)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 221)(68 222)(69 217)(70 218)(71 219)(72 220)(79 175)(80 176)(81 177)(82 178)(83 179)(84 180)(85 93)(86 94)(87 95)(88 96)(89 91)(90 92)(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)(133 166)(134 167)(135 168)(136 163)(137 164)(138 165)(139 172)(140 173)(141 174)(142 169)(143 170)(144 171)(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 17 60 50 115)(2 18 55 51 116)(3 13 56 52 117)(4 14 57 53 118)(5 15 58 54 119)(6 16 59 49 120)(7 223 64 166 172)(8 224 65 167 173)(9 225 66 168 174)(10 226 61 163 169)(11 227 62 164 170)(12 228 63 165 171)(19 162 76 88 215)(20 157 77 89 216)(21 158 78 90 211)(22 159 73 85 212)(23 160 74 86 213)(24 161 75 87 214)(25 71 111 80 32)(26 72 112 81 33)(27 67 113 82 34)(28 68 114 83 35)(29 69 109 84 36)(30 70 110 79 31)(37 131 44 92 184)(38 132 45 93 185)(39 127 46 94 186)(40 128 47 95 181)(41 129 48 96 182)(42 130 43 91 183)(97 136 142 108 193)(98 137 143 103 194)(99 138 144 104 195)(100 133 139 105 196)(101 134 140 106 197)(102 135 141 107 198)(121 239 221 200 178)(122 240 222 201 179)(123 235 217 202 180)(124 236 218 203 175)(125 237 219 204 176)(126 238 220 199 177)(145 154 206 188 233)(146 155 207 189 234)(147 156 208 190 229)(148 151 209 191 230)(149 152 210 192 231)(150 153 205 187 232)
(1 142 69 91)(2 143 70 92)(3 144 71 93)(4 139 72 94)(5 140 67 95)(6 141 68 96)(7 238 213 205)(8 239 214 206)(9 240 215 207)(10 235 216 208)(11 236 211 209)(12 237 212 210)(13 138 111 45)(14 133 112 46)(15 134 113 47)(16 135 114 48)(17 136 109 43)(18 137 110 44)(19 155 225 122)(20 156 226 123)(21 151 227 124)(22 152 228 125)(23 153 223 126)(24 154 224 121)(25 185 117 104)(26 186 118 105)(27 181 119 106)(28 182 120 107)(29 183 115 108)(30 184 116 103)(31 37 51 194)(32 38 52 195)(33 39 53 196)(34 40 54 197)(35 41 49 198)(36 42 50 193)(55 98 79 131)(56 99 80 132)(57 100 81 127)(58 101 82 128)(59 102 83 129)(60 97 84 130)(61 180 157 147)(62 175 158 148)(63 176 159 149)(64 177 160 150)(65 178 161 145)(66 179 162 146)(73 231 165 204)(74 232 166 199)(75 233 167 200)(76 234 168 201)(77 229 163 202)(78 230 164 203)(85 192 171 219)(86 187 172 220)(87 188 173 221)(88 189 174 222)(89 190 169 217)(90 191 170 218)
(1 72)(2 67)(3 68)(4 69)(5 70)(6 71)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(19 159)(20 160)(21 161)(22 162)(23 157)(24 158)(31 58)(32 59)(33 60)(34 55)(35 56)(36 57)(37 128)(38 129)(39 130)(40 131)(41 132)(42 127)(43 186)(44 181)(45 182)(46 183)(47 184)(48 185)(49 80)(50 81)(51 82)(52 83)(53 84)(54 79)(61 223)(62 224)(63 225)(64 226)(65 227)(66 228)(73 215)(74 216)(75 211)(76 212)(77 213)(78 214)(85 88)(86 89)(87 90)(91 94)(92 95)(93 96)(97 196)(98 197)(99 198)(100 193)(101 194)(102 195)(103 134)(104 135)(105 136)(106 137)(107 138)(108 133)(109 118)(110 119)(111 120)(112 115)(113 116)(114 117)(121 148)(122 149)(123 150)(124 145)(125 146)(126 147)(139 142)(140 143)(141 144)(151 178)(152 179)(153 180)(154 175)(155 176)(156 177)(169 172)(170 173)(171 174)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(199 208)(200 209)(201 210)(202 205)(203 206)(204 207)(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)(7,105)(8,106)(9,107)(10,108)(11,103)(12,104)(13,231)(14,232)(15,233)(16,234)(17,229)(18,230)(19,41)(20,42)(21,37)(22,38)(23,39)(24,40)(25,237)(26,238)(27,239)(28,240)(29,235)(30,236)(31,124)(32,125)(33,126)(34,121)(35,122)(36,123)(43,77)(44,78)(45,73)(46,74)(47,75)(48,76)(49,155)(50,156)(51,151)(52,152)(53,153)(54,154)(55,148)(56,149)(57,150)(58,145)(59,146)(60,147)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,221)(68,222)(69,217)(70,218)(71,219)(72,220)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92)(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)(133,166)(134,167)(135,168)(136,163)(137,164)(138,165)(139,172)(140,173)(141,174)(142,169)(143,170)(144,171)(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), (1,17,60,50,115)(2,18,55,51,116)(3,13,56,52,117)(4,14,57,53,118)(5,15,58,54,119)(6,16,59,49,120)(7,223,64,166,172)(8,224,65,167,173)(9,225,66,168,174)(10,226,61,163,169)(11,227,62,164,170)(12,228,63,165,171)(19,162,76,88,215)(20,157,77,89,216)(21,158,78,90,211)(22,159,73,85,212)(23,160,74,86,213)(24,161,75,87,214)(25,71,111,80,32)(26,72,112,81,33)(27,67,113,82,34)(28,68,114,83,35)(29,69,109,84,36)(30,70,110,79,31)(37,131,44,92,184)(38,132,45,93,185)(39,127,46,94,186)(40,128,47,95,181)(41,129,48,96,182)(42,130,43,91,183)(97,136,142,108,193)(98,137,143,103,194)(99,138,144,104,195)(100,133,139,105,196)(101,134,140,106,197)(102,135,141,107,198)(121,239,221,200,178)(122,240,222,201,179)(123,235,217,202,180)(124,236,218,203,175)(125,237,219,204,176)(126,238,220,199,177)(145,154,206,188,233)(146,155,207,189,234)(147,156,208,190,229)(148,151,209,191,230)(149,152,210,192,231)(150,153,205,187,232), (1,142,69,91)(2,143,70,92)(3,144,71,93)(4,139,72,94)(5,140,67,95)(6,141,68,96)(7,238,213,205)(8,239,214,206)(9,240,215,207)(10,235,216,208)(11,236,211,209)(12,237,212,210)(13,138,111,45)(14,133,112,46)(15,134,113,47)(16,135,114,48)(17,136,109,43)(18,137,110,44)(19,155,225,122)(20,156,226,123)(21,151,227,124)(22,152,228,125)(23,153,223,126)(24,154,224,121)(25,185,117,104)(26,186,118,105)(27,181,119,106)(28,182,120,107)(29,183,115,108)(30,184,116,103)(31,37,51,194)(32,38,52,195)(33,39,53,196)(34,40,54,197)(35,41,49,198)(36,42,50,193)(55,98,79,131)(56,99,80,132)(57,100,81,127)(58,101,82,128)(59,102,83,129)(60,97,84,130)(61,180,157,147)(62,175,158,148)(63,176,159,149)(64,177,160,150)(65,178,161,145)(66,179,162,146)(73,231,165,204)(74,232,166,199)(75,233,167,200)(76,234,168,201)(77,229,163,202)(78,230,164,203)(85,192,171,219)(86,187,172,220)(87,188,173,221)(88,189,174,222)(89,190,169,217)(90,191,170,218), (1,72)(2,67)(3,68)(4,69)(5,70)(6,71)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(19,159)(20,160)(21,161)(22,162)(23,157)(24,158)(31,58)(32,59)(33,60)(34,55)(35,56)(36,57)(37,128)(38,129)(39,130)(40,131)(41,132)(42,127)(43,186)(44,181)(45,182)(46,183)(47,184)(48,185)(49,80)(50,81)(51,82)(52,83)(53,84)(54,79)(61,223)(62,224)(63,225)(64,226)(65,227)(66,228)(73,215)(74,216)(75,211)(76,212)(77,213)(78,214)(85,88)(86,89)(87,90)(91,94)(92,95)(93,96)(97,196)(98,197)(99,198)(100,193)(101,194)(102,195)(103,134)(104,135)(105,136)(106,137)(107,138)(108,133)(109,118)(110,119)(111,120)(112,115)(113,116)(114,117)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(139,142)(140,143)(141,144)(151,178)(152,179)(153,180)(154,175)(155,176)(156,177)(169,172)(170,173)(171,174)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(199,208)(200,209)(201,210)(202,205)(203,206)(204,207)(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)(7,105)(8,106)(9,107)(10,108)(11,103)(12,104)(13,231)(14,232)(15,233)(16,234)(17,229)(18,230)(19,41)(20,42)(21,37)(22,38)(23,39)(24,40)(25,237)(26,238)(27,239)(28,240)(29,235)(30,236)(31,124)(32,125)(33,126)(34,121)(35,122)(36,123)(43,77)(44,78)(45,73)(46,74)(47,75)(48,76)(49,155)(50,156)(51,151)(52,152)(53,153)(54,154)(55,148)(56,149)(57,150)(58,145)(59,146)(60,147)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,221)(68,222)(69,217)(70,218)(71,219)(72,220)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92)(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)(133,166)(134,167)(135,168)(136,163)(137,164)(138,165)(139,172)(140,173)(141,174)(142,169)(143,170)(144,171)(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), (1,17,60,50,115)(2,18,55,51,116)(3,13,56,52,117)(4,14,57,53,118)(5,15,58,54,119)(6,16,59,49,120)(7,223,64,166,172)(8,224,65,167,173)(9,225,66,168,174)(10,226,61,163,169)(11,227,62,164,170)(12,228,63,165,171)(19,162,76,88,215)(20,157,77,89,216)(21,158,78,90,211)(22,159,73,85,212)(23,160,74,86,213)(24,161,75,87,214)(25,71,111,80,32)(26,72,112,81,33)(27,67,113,82,34)(28,68,114,83,35)(29,69,109,84,36)(30,70,110,79,31)(37,131,44,92,184)(38,132,45,93,185)(39,127,46,94,186)(40,128,47,95,181)(41,129,48,96,182)(42,130,43,91,183)(97,136,142,108,193)(98,137,143,103,194)(99,138,144,104,195)(100,133,139,105,196)(101,134,140,106,197)(102,135,141,107,198)(121,239,221,200,178)(122,240,222,201,179)(123,235,217,202,180)(124,236,218,203,175)(125,237,219,204,176)(126,238,220,199,177)(145,154,206,188,233)(146,155,207,189,234)(147,156,208,190,229)(148,151,209,191,230)(149,152,210,192,231)(150,153,205,187,232), (1,142,69,91)(2,143,70,92)(3,144,71,93)(4,139,72,94)(5,140,67,95)(6,141,68,96)(7,238,213,205)(8,239,214,206)(9,240,215,207)(10,235,216,208)(11,236,211,209)(12,237,212,210)(13,138,111,45)(14,133,112,46)(15,134,113,47)(16,135,114,48)(17,136,109,43)(18,137,110,44)(19,155,225,122)(20,156,226,123)(21,151,227,124)(22,152,228,125)(23,153,223,126)(24,154,224,121)(25,185,117,104)(26,186,118,105)(27,181,119,106)(28,182,120,107)(29,183,115,108)(30,184,116,103)(31,37,51,194)(32,38,52,195)(33,39,53,196)(34,40,54,197)(35,41,49,198)(36,42,50,193)(55,98,79,131)(56,99,80,132)(57,100,81,127)(58,101,82,128)(59,102,83,129)(60,97,84,130)(61,180,157,147)(62,175,158,148)(63,176,159,149)(64,177,160,150)(65,178,161,145)(66,179,162,146)(73,231,165,204)(74,232,166,199)(75,233,167,200)(76,234,168,201)(77,229,163,202)(78,230,164,203)(85,192,171,219)(86,187,172,220)(87,188,173,221)(88,189,174,222)(89,190,169,217)(90,191,170,218), (1,72)(2,67)(3,68)(4,69)(5,70)(6,71)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(19,159)(20,160)(21,161)(22,162)(23,157)(24,158)(31,58)(32,59)(33,60)(34,55)(35,56)(36,57)(37,128)(38,129)(39,130)(40,131)(41,132)(42,127)(43,186)(44,181)(45,182)(46,183)(47,184)(48,185)(49,80)(50,81)(51,82)(52,83)(53,84)(54,79)(61,223)(62,224)(63,225)(64,226)(65,227)(66,228)(73,215)(74,216)(75,211)(76,212)(77,213)(78,214)(85,88)(86,89)(87,90)(91,94)(92,95)(93,96)(97,196)(98,197)(99,198)(100,193)(101,194)(102,195)(103,134)(104,135)(105,136)(106,137)(107,138)(108,133)(109,118)(110,119)(111,120)(112,115)(113,116)(114,117)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(139,142)(140,143)(141,144)(151,178)(152,179)(153,180)(154,175)(155,176)(156,177)(169,172)(170,173)(171,174)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(199,208)(200,209)(201,210)(202,205)(203,206)(204,207)(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),(7,105),(8,106),(9,107),(10,108),(11,103),(12,104),(13,231),(14,232),(15,233),(16,234),(17,229),(18,230),(19,41),(20,42),(21,37),(22,38),(23,39),(24,40),(25,237),(26,238),(27,239),(28,240),(29,235),(30,236),(31,124),(32,125),(33,126),(34,121),(35,122),(36,123),(43,77),(44,78),(45,73),(46,74),(47,75),(48,76),(49,155),(50,156),(51,151),(52,152),(53,153),(54,154),(55,148),(56,149),(57,150),(58,145),(59,146),(60,147),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,221),(68,222),(69,217),(70,218),(71,219),(72,220),(79,175),(80,176),(81,177),(82,178),(83,179),(84,180),(85,93),(86,94),(87,95),(88,96),(89,91),(90,92),(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),(133,166),(134,167),(135,168),(136,163),(137,164),(138,165),(139,172),(140,173),(141,174),(142,169),(143,170),(144,171),(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)], [(1,17,60,50,115),(2,18,55,51,116),(3,13,56,52,117),(4,14,57,53,118),(5,15,58,54,119),(6,16,59,49,120),(7,223,64,166,172),(8,224,65,167,173),(9,225,66,168,174),(10,226,61,163,169),(11,227,62,164,170),(12,228,63,165,171),(19,162,76,88,215),(20,157,77,89,216),(21,158,78,90,211),(22,159,73,85,212),(23,160,74,86,213),(24,161,75,87,214),(25,71,111,80,32),(26,72,112,81,33),(27,67,113,82,34),(28,68,114,83,35),(29,69,109,84,36),(30,70,110,79,31),(37,131,44,92,184),(38,132,45,93,185),(39,127,46,94,186),(40,128,47,95,181),(41,129,48,96,182),(42,130,43,91,183),(97,136,142,108,193),(98,137,143,103,194),(99,138,144,104,195),(100,133,139,105,196),(101,134,140,106,197),(102,135,141,107,198),(121,239,221,200,178),(122,240,222,201,179),(123,235,217,202,180),(124,236,218,203,175),(125,237,219,204,176),(126,238,220,199,177),(145,154,206,188,233),(146,155,207,189,234),(147,156,208,190,229),(148,151,209,191,230),(149,152,210,192,231),(150,153,205,187,232)], [(1,142,69,91),(2,143,70,92),(3,144,71,93),(4,139,72,94),(5,140,67,95),(6,141,68,96),(7,238,213,205),(8,239,214,206),(9,240,215,207),(10,235,216,208),(11,236,211,209),(12,237,212,210),(13,138,111,45),(14,133,112,46),(15,134,113,47),(16,135,114,48),(17,136,109,43),(18,137,110,44),(19,155,225,122),(20,156,226,123),(21,151,227,124),(22,152,228,125),(23,153,223,126),(24,154,224,121),(25,185,117,104),(26,186,118,105),(27,181,119,106),(28,182,120,107),(29,183,115,108),(30,184,116,103),(31,37,51,194),(32,38,52,195),(33,39,53,196),(34,40,54,197),(35,41,49,198),(36,42,50,193),(55,98,79,131),(56,99,80,132),(57,100,81,127),(58,101,82,128),(59,102,83,129),(60,97,84,130),(61,180,157,147),(62,175,158,148),(63,176,159,149),(64,177,160,150),(65,178,161,145),(66,179,162,146),(73,231,165,204),(74,232,166,199),(75,233,167,200),(76,234,168,201),(77,229,163,202),(78,230,164,203),(85,192,171,219),(86,187,172,220),(87,188,173,221),(88,189,174,222),(89,190,169,217),(90,191,170,218)], [(1,72),(2,67),(3,68),(4,69),(5,70),(6,71),(7,163),(8,164),(9,165),(10,166),(11,167),(12,168),(13,28),(14,29),(15,30),(16,25),(17,26),(18,27),(19,159),(20,160),(21,161),(22,162),(23,157),(24,158),(31,58),(32,59),(33,60),(34,55),(35,56),(36,57),(37,128),(38,129),(39,130),(40,131),(41,132),(42,127),(43,186),(44,181),(45,182),(46,183),(47,184),(48,185),(49,80),(50,81),(51,82),(52,83),(53,84),(54,79),(61,223),(62,224),(63,225),(64,226),(65,227),(66,228),(73,215),(74,216),(75,211),(76,212),(77,213),(78,214),(85,88),(86,89),(87,90),(91,94),(92,95),(93,96),(97,196),(98,197),(99,198),(100,193),(101,194),(102,195),(103,134),(104,135),(105,136),(106,137),(107,138),(108,133),(109,118),(110,119),(111,120),(112,115),(113,116),(114,117),(121,148),(122,149),(123,150),(124,145),(125,146),(126,147),(139,142),(140,143),(141,144),(151,178),(152,179),(153,180),(154,175),(155,176),(156,177),(169,172),(170,173),(171,174),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222),(199,208),(200,209),(201,210),(202,205),(203,206),(204,207),(229,238),(230,239),(231,240),(232,235),(233,236),(234,237)])
Matrix representation ►G ⊆ 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] >;
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 |
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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