direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C23.26D6, (C2×C60)⋊25C4, (C2×C12)⋊6C20, C60.256(C2×C4), C12.42(C2×C20), C4⋊Dic3⋊17C10, (C2×C20)⋊14Dic3, (C2×C20).439D6, (C4×Dic3)⋊15C10, (Dic3×C20)⋊33C2, C6.24(C22×C20), (C22×C60).26C2, (C22×C20).21S3, C23.26(S3×C10), C20.70(C2×Dic3), C4.15(C10×Dic3), C15⋊32(C42⋊C2), C30.209(C4○D4), (C2×C30).423C23, (C22×C12).10C10, C30.231(C22×C4), (C2×C60).568C22, C6.D4.5C10, (C22×C10).123D6, C10.123(C4○D12), C22.5(C10×Dic3), C10.47(C22×Dic3), (C22×C30).174C22, (C10×Dic3).225C22, C3⋊4(C5×C42⋊C2), (C2×C4)⋊4(C5×Dic3), C6.14(C5×C4○D4), C2.4(C5×C4○D12), C2.5(Dic3×C2×C10), (C2×C6).35(C2×C20), (C5×C4⋊Dic3)⋊35C2, (C22×C4).9(C5×S3), C22.22(S3×C2×C10), (C2×C4).102(S3×C10), (C2×C30).203(C2×C4), (C2×C12).98(C2×C10), (C2×C6).44(C22×C10), (C22×C6).36(C2×C10), (C2×C10).44(C2×Dic3), (C2×C10).357(C22×S3), (C2×Dic3).33(C2×C10), (C5×C6.D4).11C2, SmallGroup(480,805)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.26D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 260 in 152 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C30, C30, C30, C42⋊C2, C2×C20, C2×C20, C2×C20, C22×C10, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C5×Dic3, C60, C2×C30, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C23.26D6, C10×Dic3, C2×C60, C2×C60, C22×C30, C5×C42⋊C2, Dic3×C20, C5×C4⋊Dic3, C5×C6.D4, C22×C60, C5×C23.26D6
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, Dic3, D6, C22×C4, C4○D4, C20, C2×C10, C2×Dic3, C22×S3, C5×S3, C42⋊C2, C2×C20, C22×C10, C4○D12, C22×Dic3, C5×Dic3, S3×C10, C22×C20, C5×C4○D4, C23.26D6, C10×Dic3, S3×C2×C10, C5×C42⋊C2, C5×C4○D12, Dic3×C2×C10, C5×C23.26D6
►On 240 points
Generators in S240
(1 50 200 18 136)(2 51 201 19 137)(3 52 202 20 138)(4 53 203 21 139)(5 54 204 22 140)(6 55 193 23 141)(7 56 194 24 142)(8 57 195 13 143)(9 58 196 14 144)(10 59 197 15 133)(11 60 198 16 134)(12 49 199 17 135)(25 229 91 97 189)(26 230 92 98 190)(27 231 93 99 191)(28 232 94 100 192)(29 233 95 101 181)(30 234 96 102 182)(31 235 85 103 183)(32 236 86 104 184)(33 237 87 105 185)(34 238 88 106 186)(35 239 89 107 187)(36 240 90 108 188)(37 125 67 164 175)(38 126 68 165 176)(39 127 69 166 177)(40 128 70 167 178)(41 129 71 168 179)(42 130 72 157 180)(43 131 61 158 169)(44 132 62 159 170)(45 121 63 160 171)(46 122 64 161 172)(47 123 65 162 173)(48 124 66 163 174)(73 118 215 224 147)(74 119 216 225 148)(75 120 205 226 149)(76 109 206 227 150)(77 110 207 228 151)(78 111 208 217 152)(79 112 209 218 153)(80 113 210 219 154)(81 114 211 220 155)(82 115 212 221 156)(83 116 213 222 145)(84 117 214 223 146)
(1 230)(2 231)(3 232)(4 233)(5 234)(6 235)(7 236)(8 237)(9 238)(10 239)(11 240)(12 229)(13 185)(14 186)(15 187)(16 188)(17 189)(18 190)(19 191)(20 192)(21 181)(22 182)(23 183)(24 184)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 133)(36 134)(37 219)(38 220)(39 221)(40 222)(41 223)(42 224)(43 225)(44 226)(45 227)(46 228)(47 217)(48 218)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 73)(97 199)(98 200)(99 201)(100 202)(101 203)(102 204)(103 193)(104 194)(105 195)(106 196)(107 197)(108 198)(109 160)(110 161)(111 162)(112 163)(113 164)(114 165)(115 166)(116 167)(117 168)(118 157)(119 158)(120 159)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 145)(129 146)(130 147)(131 148)(132 149)(169 216)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 211)(177 212)(178 213)(179 214)(180 215)
(1 230)(2 231)(3 232)(4 233)(5 234)(6 235)(7 236)(8 237)(9 238)(10 239)(11 240)(12 229)(13 185)(14 186)(15 187)(16 188)(17 189)(18 190)(19 191)(20 192)(21 181)(22 182)(23 183)(24 184)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 133)(36 134)(37 225)(38 226)(39 227)(40 228)(41 217)(42 218)(43 219)(44 220)(45 221)(46 222)(47 223)(48 224)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)(97 199)(98 200)(99 201)(100 202)(101 203)(102 204)(103 193)(104 194)(105 195)(106 196)(107 197)(108 198)(109 166)(110 167)(111 168)(112 157)(113 158)(114 159)(115 160)(116 161)(117 162)(118 163)(119 164)(120 165)(121 156)(122 145)(123 146)(124 147)(125 148)(126 149)(127 150)(128 151)(129 152)(130 153)(131 154)(132 155)(169 210)(170 211)(171 212)(172 213)(173 214)(174 215)(175 216)(176 205)(177 206)(178 207)(179 208)(180 209)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)(193 199)(194 200)(195 201)(196 202)(197 203)(198 204)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)(217 223)(218 224)(219 225)(220 226)(221 227)(222 228)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)
(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)
(1 180 236 215)(2 173 237 208)(3 178 238 213)(4 171 239 206)(5 176 240 211)(6 169 229 216)(7 174 230 209)(8 179 231 214)(9 172 232 207)(10 177 233 212)(11 170 234 205)(12 175 235 210)(13 71 191 84)(14 64 192 77)(15 69 181 82)(16 62 182 75)(17 67 183 80)(18 72 184 73)(19 65 185 78)(20 70 186 83)(21 63 187 76)(22 68 188 81)(23 61 189 74)(24 66 190 79)(25 119 141 158)(26 112 142 163)(27 117 143 168)(28 110 144 161)(29 115 133 166)(30 120 134 159)(31 113 135 164)(32 118 136 157)(33 111 137 162)(34 116 138 167)(35 109 139 160)(36 114 140 165)(37 85 219 49)(38 90 220 54)(39 95 221 59)(40 88 222 52)(41 93 223 57)(42 86 224 50)(43 91 225 55)(44 96 226 60)(45 89 227 53)(46 94 228 58)(47 87 217 51)(48 92 218 56)(97 148 193 131)(98 153 194 124)(99 146 195 129)(100 151 196 122)(101 156 197 127)(102 149 198 132)(103 154 199 125)(104 147 200 130)(105 152 201 123)(106 145 202 128)(107 150 203 121)(108 155 204 126)
G:=sub<Sym(240)| (1,50,200,18,136)(2,51,201,19,137)(3,52,202,20,138)(4,53,203,21,139)(5,54,204,22,140)(6,55,193,23,141)(7,56,194,24,142)(8,57,195,13,143)(9,58,196,14,144)(10,59,197,15,133)(11,60,198,16,134)(12,49,199,17,135)(25,229,91,97,189)(26,230,92,98,190)(27,231,93,99,191)(28,232,94,100,192)(29,233,95,101,181)(30,234,96,102,182)(31,235,85,103,183)(32,236,86,104,184)(33,237,87,105,185)(34,238,88,106,186)(35,239,89,107,187)(36,240,90,108,188)(37,125,67,164,175)(38,126,68,165,176)(39,127,69,166,177)(40,128,70,167,178)(41,129,71,168,179)(42,130,72,157,180)(43,131,61,158,169)(44,132,62,159,170)(45,121,63,160,171)(46,122,64,161,172)(47,123,65,162,173)(48,124,66,163,174)(73,118,215,224,147)(74,119,216,225,148)(75,120,205,226,149)(76,109,206,227,150)(77,110,207,228,151)(78,111,208,217,152)(79,112,209,218,153)(80,113,210,219,154)(81,114,211,220,155)(82,115,212,221,156)(83,116,213,222,145)(84,117,214,223,146), (1,230)(2,231)(3,232)(4,233)(5,234)(6,235)(7,236)(8,237)(9,238)(10,239)(11,240)(12,229)(13,185)(14,186)(15,187)(16,188)(17,189)(18,190)(19,191)(20,192)(21,181)(22,182)(23,183)(24,184)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,219)(38,220)(39,221)(40,222)(41,223)(42,224)(43,225)(44,226)(45,227)(46,228)(47,217)(48,218)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,73)(97,199)(98,200)(99,201)(100,202)(101,203)(102,204)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,160)(110,161)(111,162)(112,163)(113,164)(114,165)(115,166)(116,167)(117,168)(118,157)(119,158)(120,159)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,145)(129,146)(130,147)(131,148)(132,149)(169,216)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,211)(177,212)(178,213)(179,214)(180,215), (1,230)(2,231)(3,232)(4,233)(5,234)(6,235)(7,236)(8,237)(9,238)(10,239)(11,240)(12,229)(13,185)(14,186)(15,187)(16,188)(17,189)(18,190)(19,191)(20,192)(21,181)(22,182)(23,183)(24,184)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,225)(38,226)(39,227)(40,228)(41,217)(42,218)(43,219)(44,220)(45,221)(46,222)(47,223)(48,224)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79)(97,199)(98,200)(99,201)(100,202)(101,203)(102,204)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,166)(110,167)(111,168)(112,157)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,156)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,205)(177,206)(178,207)(179,208)(180,209), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (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), (1,180,236,215)(2,173,237,208)(3,178,238,213)(4,171,239,206)(5,176,240,211)(6,169,229,216)(7,174,230,209)(8,179,231,214)(9,172,232,207)(10,177,233,212)(11,170,234,205)(12,175,235,210)(13,71,191,84)(14,64,192,77)(15,69,181,82)(16,62,182,75)(17,67,183,80)(18,72,184,73)(19,65,185,78)(20,70,186,83)(21,63,187,76)(22,68,188,81)(23,61,189,74)(24,66,190,79)(25,119,141,158)(26,112,142,163)(27,117,143,168)(28,110,144,161)(29,115,133,166)(30,120,134,159)(31,113,135,164)(32,118,136,157)(33,111,137,162)(34,116,138,167)(35,109,139,160)(36,114,140,165)(37,85,219,49)(38,90,220,54)(39,95,221,59)(40,88,222,52)(41,93,223,57)(42,86,224,50)(43,91,225,55)(44,96,226,60)(45,89,227,53)(46,94,228,58)(47,87,217,51)(48,92,218,56)(97,148,193,131)(98,153,194,124)(99,146,195,129)(100,151,196,122)(101,156,197,127)(102,149,198,132)(103,154,199,125)(104,147,200,130)(105,152,201,123)(106,145,202,128)(107,150,203,121)(108,155,204,126)>;
G:=Group( (1,50,200,18,136)(2,51,201,19,137)(3,52,202,20,138)(4,53,203,21,139)(5,54,204,22,140)(6,55,193,23,141)(7,56,194,24,142)(8,57,195,13,143)(9,58,196,14,144)(10,59,197,15,133)(11,60,198,16,134)(12,49,199,17,135)(25,229,91,97,189)(26,230,92,98,190)(27,231,93,99,191)(28,232,94,100,192)(29,233,95,101,181)(30,234,96,102,182)(31,235,85,103,183)(32,236,86,104,184)(33,237,87,105,185)(34,238,88,106,186)(35,239,89,107,187)(36,240,90,108,188)(37,125,67,164,175)(38,126,68,165,176)(39,127,69,166,177)(40,128,70,167,178)(41,129,71,168,179)(42,130,72,157,180)(43,131,61,158,169)(44,132,62,159,170)(45,121,63,160,171)(46,122,64,161,172)(47,123,65,162,173)(48,124,66,163,174)(73,118,215,224,147)(74,119,216,225,148)(75,120,205,226,149)(76,109,206,227,150)(77,110,207,228,151)(78,111,208,217,152)(79,112,209,218,153)(80,113,210,219,154)(81,114,211,220,155)(82,115,212,221,156)(83,116,213,222,145)(84,117,214,223,146), (1,230)(2,231)(3,232)(4,233)(5,234)(6,235)(7,236)(8,237)(9,238)(10,239)(11,240)(12,229)(13,185)(14,186)(15,187)(16,188)(17,189)(18,190)(19,191)(20,192)(21,181)(22,182)(23,183)(24,184)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,219)(38,220)(39,221)(40,222)(41,223)(42,224)(43,225)(44,226)(45,227)(46,228)(47,217)(48,218)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,73)(97,199)(98,200)(99,201)(100,202)(101,203)(102,204)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,160)(110,161)(111,162)(112,163)(113,164)(114,165)(115,166)(116,167)(117,168)(118,157)(119,158)(120,159)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,145)(129,146)(130,147)(131,148)(132,149)(169,216)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,211)(177,212)(178,213)(179,214)(180,215), (1,230)(2,231)(3,232)(4,233)(5,234)(6,235)(7,236)(8,237)(9,238)(10,239)(11,240)(12,229)(13,185)(14,186)(15,187)(16,188)(17,189)(18,190)(19,191)(20,192)(21,181)(22,182)(23,183)(24,184)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,225)(38,226)(39,227)(40,228)(41,217)(42,218)(43,219)(44,220)(45,221)(46,222)(47,223)(48,224)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79)(97,199)(98,200)(99,201)(100,202)(101,203)(102,204)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,166)(110,167)(111,168)(112,157)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,156)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,205)(177,206)(178,207)(179,208)(180,209), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (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), (1,180,236,215)(2,173,237,208)(3,178,238,213)(4,171,239,206)(5,176,240,211)(6,169,229,216)(7,174,230,209)(8,179,231,214)(9,172,232,207)(10,177,233,212)(11,170,234,205)(12,175,235,210)(13,71,191,84)(14,64,192,77)(15,69,181,82)(16,62,182,75)(17,67,183,80)(18,72,184,73)(19,65,185,78)(20,70,186,83)(21,63,187,76)(22,68,188,81)(23,61,189,74)(24,66,190,79)(25,119,141,158)(26,112,142,163)(27,117,143,168)(28,110,144,161)(29,115,133,166)(30,120,134,159)(31,113,135,164)(32,118,136,157)(33,111,137,162)(34,116,138,167)(35,109,139,160)(36,114,140,165)(37,85,219,49)(38,90,220,54)(39,95,221,59)(40,88,222,52)(41,93,223,57)(42,86,224,50)(43,91,225,55)(44,96,226,60)(45,89,227,53)(46,94,228,58)(47,87,217,51)(48,92,218,56)(97,148,193,131)(98,153,194,124)(99,146,195,129)(100,151,196,122)(101,156,197,127)(102,149,198,132)(103,154,199,125)(104,147,200,130)(105,152,201,123)(106,145,202,128)(107,150,203,121)(108,155,204,126) );
G=PermutationGroup([[(1,50,200,18,136),(2,51,201,19,137),(3,52,202,20,138),(4,53,203,21,139),(5,54,204,22,140),(6,55,193,23,141),(7,56,194,24,142),(8,57,195,13,143),(9,58,196,14,144),(10,59,197,15,133),(11,60,198,16,134),(12,49,199,17,135),(25,229,91,97,189),(26,230,92,98,190),(27,231,93,99,191),(28,232,94,100,192),(29,233,95,101,181),(30,234,96,102,182),(31,235,85,103,183),(32,236,86,104,184),(33,237,87,105,185),(34,238,88,106,186),(35,239,89,107,187),(36,240,90,108,188),(37,125,67,164,175),(38,126,68,165,176),(39,127,69,166,177),(40,128,70,167,178),(41,129,71,168,179),(42,130,72,157,180),(43,131,61,158,169),(44,132,62,159,170),(45,121,63,160,171),(46,122,64,161,172),(47,123,65,162,173),(48,124,66,163,174),(73,118,215,224,147),(74,119,216,225,148),(75,120,205,226,149),(76,109,206,227,150),(77,110,207,228,151),(78,111,208,217,152),(79,112,209,218,153),(80,113,210,219,154),(81,114,211,220,155),(82,115,212,221,156),(83,116,213,222,145),(84,117,214,223,146)], [(1,230),(2,231),(3,232),(4,233),(5,234),(6,235),(7,236),(8,237),(9,238),(10,239),(11,240),(12,229),(13,185),(14,186),(15,187),(16,188),(17,189),(18,190),(19,191),(20,192),(21,181),(22,182),(23,183),(24,184),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,133),(36,134),(37,219),(38,220),(39,221),(40,222),(41,223),(42,224),(43,225),(44,226),(45,227),(46,228),(47,217),(48,218),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,73),(97,199),(98,200),(99,201),(100,202),(101,203),(102,204),(103,193),(104,194),(105,195),(106,196),(107,197),(108,198),(109,160),(110,161),(111,162),(112,163),(113,164),(114,165),(115,166),(116,167),(117,168),(118,157),(119,158),(120,159),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,145),(129,146),(130,147),(131,148),(132,149),(169,216),(170,205),(171,206),(172,207),(173,208),(174,209),(175,210),(176,211),(177,212),(178,213),(179,214),(180,215)], [(1,230),(2,231),(3,232),(4,233),(5,234),(6,235),(7,236),(8,237),(9,238),(10,239),(11,240),(12,229),(13,185),(14,186),(15,187),(16,188),(17,189),(18,190),(19,191),(20,192),(21,181),(22,182),(23,183),(24,184),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,133),(36,134),(37,225),(38,226),(39,227),(40,228),(41,217),(42,218),(43,219),(44,220),(45,221),(46,222),(47,223),(48,224),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,80),(62,81),(63,82),(64,83),(65,84),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79),(97,199),(98,200),(99,201),(100,202),(101,203),(102,204),(103,193),(104,194),(105,195),(106,196),(107,197),(108,198),(109,166),(110,167),(111,168),(112,157),(113,158),(114,159),(115,160),(116,161),(117,162),(118,163),(119,164),(120,165),(121,156),(122,145),(123,146),(124,147),(125,148),(126,149),(127,150),(128,151),(129,152),(130,153),(131,154),(132,155),(169,210),(170,211),(171,212),(172,213),(173,214),(174,215),(175,216),(176,205),(177,206),(178,207),(179,208),(180,209)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132),(133,139),(134,140),(135,141),(136,142),(137,143),(138,144),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156),(157,163),(158,164),(159,165),(160,166),(161,167),(162,168),(169,175),(170,176),(171,177),(172,178),(173,179),(174,180),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192),(193,199),(194,200),(195,201),(196,202),(197,203),(198,204),(205,211),(206,212),(207,213),(208,214),(209,215),(210,216),(217,223),(218,224),(219,225),(220,226),(221,227),(222,228),(229,235),(230,236),(231,237),(232,238),(233,239),(234,240)], [(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)], [(1,180,236,215),(2,173,237,208),(3,178,238,213),(4,171,239,206),(5,176,240,211),(6,169,229,216),(7,174,230,209),(8,179,231,214),(9,172,232,207),(10,177,233,212),(11,170,234,205),(12,175,235,210),(13,71,191,84),(14,64,192,77),(15,69,181,82),(16,62,182,75),(17,67,183,80),(18,72,184,73),(19,65,185,78),(20,70,186,83),(21,63,187,76),(22,68,188,81),(23,61,189,74),(24,66,190,79),(25,119,141,158),(26,112,142,163),(27,117,143,168),(28,110,144,161),(29,115,133,166),(30,120,134,159),(31,113,135,164),(32,118,136,157),(33,111,137,162),(34,116,138,167),(35,109,139,160),(36,114,140,165),(37,85,219,49),(38,90,220,54),(39,95,221,59),(40,88,222,52),(41,93,223,57),(42,86,224,50),(43,91,225,55),(44,96,226,60),(45,89,227,53),(46,94,228,58),(47,87,217,51),(48,92,218,56),(97,148,193,131),(98,153,194,124),(99,146,195,129),(100,151,196,122),(101,156,197,127),(102,149,198,132),(103,154,199,125),(104,147,200,130),(105,152,201,123),(106,145,202,128),(107,150,203,121),(108,155,204,126)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20BD | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | S3 | Dic3 | D6 | D6 | C4○D4 | C5×S3 | C4○D12 | C5×Dic3 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C4○D12 |
| kernel | C5×C23.26D6 | Dic3×C20 | C5×C4⋊Dic3 | C5×C6.D4 | C22×C60 | C2×C60 | C23.26D6 | C4×Dic3 | C4⋊Dic3 | C6.D4 | C22×C12 | C2×C12 | C22×C20 | C2×C20 | C2×C20 | C22×C10 | C30 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 8 | 8 | 8 | 4 | 32 | 1 | 4 | 2 | 1 | 4 | 4 | 8 | 16 | 8 | 4 | 16 | 32 |
Matrix representation of C5×C23.26D6 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 60 | 0 | 0 | 0 |
| 50 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 21 | 0 |
| 0 | 0 | 0 | 29 |
| 60 | 39 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 29 |
| 0 | 0 | 21 | 0 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[60,50,0,0,0,1,0,0,0,0,1,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[11,0,0,0,0,11,0,0,0,0,21,0,0,0,0,29],[60,0,0,0,39,1,0,0,0,0,0,21,0,0,29,0] >;
C5×C23.26D6 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{26}D_6 % in TeX
G:=Group("C5xC2^3.26D6"); // GroupNames label
G:=SmallGroup(480,805);
// by ID
G=gap.SmallGroup(480,805);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,568,1766,15686]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^6=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations