direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×S3×C5⋊2C8, C60.178C23, C10⋊5(S3×C8), C30⋊6(C2×C8), (S3×C10)⋊4C8, C15⋊8(C22×C8), (S3×C20).10C4, C20.110(C4×S3), C60.148(C2×C4), (C4×S3).48D10, (C2×C20).328D6, (C4×S3).5Dic5, D6.6(C2×Dic5), C4.22(S3×Dic5), C15⋊3C8⋊43C22, (C2×C12).332D10, C12.27(C2×Dic5), C6.1(C22×Dic5), (S3×C20).53C22, C20.175(C22×S3), (C2×C60).230C22, C30.100(C22×C4), (C2×Dic3).8Dic5, Dic3.9(C2×Dic5), (C10×Dic3).16C4, C12.175(C22×D5), (C22×S3).5Dic5, C22.13(S3×Dic5), C5⋊7(S3×C2×C8), C6⋊1(C2×C5⋊2C8), (C5×S3)⋊4(C2×C8), (S3×C2×C10).9C4, (S3×C2×C20).8C2, C3⋊1(C22×C5⋊2C8), (S3×C2×C4).12D5, (C6×C5⋊2C8)⋊11C2, C4.148(C2×S3×D5), C2.1(C2×S3×Dic5), C10.109(S3×C2×C4), (C2×C15⋊3C8)⋊24C2, (C2×C30).97(C2×C4), (C2×C10).74(C4×S3), (C2×C4).233(S3×D5), (S3×C10).33(C2×C4), (C3×C5⋊2C8)⋊34C22, (C2×C6).14(C2×Dic5), (C5×Dic3).41(C2×C4), SmallGroup(480,361)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C2×S3×C5⋊2C8 |
Generators and relations for C2×S3×C5⋊2C8
G = < a,b,c,d,e | a2=b3=c2=d5=e8=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 412 in 152 conjugacy classes, 84 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C22×C8, C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C22×C10, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C60, S3×C10, C2×C30, C2×C5⋊2C8, C2×C5⋊2C8, C22×C20, S3×C2×C8, C3×C5⋊2C8, C15⋊3C8, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C22×C5⋊2C8, S3×C5⋊2C8, C6×C5⋊2C8, C2×C15⋊3C8, S3×C2×C20, C2×S3×C5⋊2C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D5, D6, C2×C8, C22×C4, Dic5, D10, C4×S3, C22×S3, C22×C8, C5⋊2C8, C2×Dic5, C22×D5, S3×C8, S3×C2×C4, S3×D5, C2×C5⋊2C8, C22×Dic5, S3×C2×C8, S3×Dic5, C2×S3×D5, C22×C5⋊2C8, S3×C5⋊2C8, C2×S3×Dic5, C2×S3×C5⋊2C8
►On 240 points
Generators in S240
(1 176)(2 169)(3 170)(4 171)(5 172)(6 173)(7 174)(8 175)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 134)(34 135)(35 136)(36 129)(37 130)(38 131)(39 132)(40 133)(41 224)(42 217)(43 218)(44 219)(45 220)(46 221)(47 222)(48 223)(57 148)(58 149)(59 150)(60 151)(61 152)(62 145)(63 146)(64 147)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 202)(74 203)(75 204)(76 205)(77 206)(78 207)(79 208)(80 201)(81 199)(82 200)(83 193)(84 194)(85 195)(86 196)(87 197)(88 198)(89 230)(90 231)(91 232)(92 225)(93 226)(94 227)(95 228)(96 229)(97 118)(98 119)(99 120)(100 113)(101 114)(102 115)(103 116)(104 117)(105 191)(106 192)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)(137 210)(138 211)(139 212)(140 213)(141 214)(142 215)(143 216)(144 209)(153 168)(154 161)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(177 235)(178 236)(179 237)(180 238)(181 239)(182 240)(183 233)(184 234)
(1 68 119)(2 69 120)(3 70 113)(4 71 114)(5 72 115)(6 65 116)(7 66 117)(8 67 118)(9 88 188)(10 81 189)(11 82 190)(12 83 191)(13 84 192)(14 85 185)(15 86 186)(16 87 187)(17 62 79)(18 63 80)(19 64 73)(20 57 74)(21 58 75)(22 59 76)(23 60 77)(24 61 78)(25 147 202)(26 148 203)(27 149 204)(28 150 205)(29 151 206)(30 152 207)(31 145 208)(32 146 201)(33 214 159)(34 215 160)(35 216 153)(36 209 154)(37 210 155)(38 211 156)(39 212 157)(40 213 158)(41 233 89)(42 234 90)(43 235 91)(44 236 92)(45 237 93)(46 238 94)(47 239 95)(48 240 96)(49 194 106)(50 195 107)(51 196 108)(52 197 109)(53 198 110)(54 199 111)(55 200 112)(56 193 105)(97 175 123)(98 176 124)(99 169 125)(100 170 126)(101 171 127)(102 172 128)(103 173 121)(104 174 122)(129 144 161)(130 137 162)(131 138 163)(132 139 164)(133 140 165)(134 141 166)(135 142 167)(136 143 168)(177 232 218)(178 225 219)(179 226 220)(180 227 221)(181 228 222)(182 229 223)(183 230 224)(184 231 217)
(1 172)(2 173)(3 174)(4 175)(5 176)(6 169)(7 170)(8 171)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 105)(17 149)(18 150)(19 151)(20 152)(21 145)(22 146)(23 147)(24 148)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 129)(41 226)(42 227)(43 228)(44 229)(45 230)(46 231)(47 232)(48 225)(49 188)(50 189)(51 190)(52 191)(53 192)(54 185)(55 186)(56 187)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 97)(72 98)(73 206)(74 207)(75 208)(76 201)(77 202)(78 203)(79 204)(80 205)(81 195)(82 196)(83 197)(84 198)(85 199)(86 200)(87 193)(88 194)(89 220)(90 221)(91 222)(92 223)(93 224)(94 217)(95 218)(96 219)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 121)(137 159)(138 160)(139 153)(140 154)(141 155)(142 156)(143 157)(144 158)(161 213)(162 214)(163 215)(164 216)(165 209)(166 210)(167 211)(168 212)(177 239)(178 240)(179 233)(180 234)(181 235)(182 236)(183 237)(184 238)
(1 78 194 183 40)(2 33 184 195 79)(3 80 196 177 34)(4 35 178 197 73)(5 74 198 179 36)(6 37 180 199 75)(7 76 200 181 38)(8 39 182 193 77)(9 45 161 102 148)(10 149 103 162 46)(11 47 163 104 150)(12 151 97 164 48)(13 41 165 98 152)(14 145 99 166 42)(15 43 167 100 146)(16 147 101 168 44)(17 69 214 231 107)(18 108 232 215 70)(19 71 216 225 109)(20 110 226 209 72)(21 65 210 227 111)(22 112 228 211 66)(23 67 212 229 105)(24 106 230 213 68)(25 127 143 92 187)(26 188 93 144 128)(27 121 137 94 189)(28 190 95 138 122)(29 123 139 96 191)(30 192 89 140 124)(31 125 141 90 185)(32 186 91 142 126)(49 224 158 119 61)(50 62 120 159 217)(51 218 160 113 63)(52 64 114 153 219)(53 220 154 115 57)(54 58 116 155 221)(55 222 156 117 59)(56 60 118 157 223)(81 204 173 130 238)(82 239 131 174 205)(83 206 175 132 240)(84 233 133 176 207)(85 208 169 134 234)(86 235 135 170 201)(87 202 171 136 236)(88 237 129 172 203)
(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)
G:=sub<Sym(240)| (1,176)(2,169)(3,170)(4,171)(5,172)(6,173)(7,174)(8,175)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,134)(34,135)(35,136)(36,129)(37,130)(38,131)(39,132)(40,133)(41,224)(42,217)(43,218)(44,219)(45,220)(46,221)(47,222)(48,223)(57,148)(58,149)(59,150)(60,151)(61,152)(62,145)(63,146)(64,147)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,202)(74,203)(75,204)(76,205)(77,206)(78,207)(79,208)(80,201)(81,199)(82,200)(83,193)(84,194)(85,195)(86,196)(87,197)(88,198)(89,230)(90,231)(91,232)(92,225)(93,226)(94,227)(95,228)(96,229)(97,118)(98,119)(99,120)(100,113)(101,114)(102,115)(103,116)(104,117)(105,191)(106,192)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(137,210)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,209)(153,168)(154,161)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(177,235)(178,236)(179,237)(180,238)(181,239)(182,240)(183,233)(184,234), (1,68,119)(2,69,120)(3,70,113)(4,71,114)(5,72,115)(6,65,116)(7,66,117)(8,67,118)(9,88,188)(10,81,189)(11,82,190)(12,83,191)(13,84,192)(14,85,185)(15,86,186)(16,87,187)(17,62,79)(18,63,80)(19,64,73)(20,57,74)(21,58,75)(22,59,76)(23,60,77)(24,61,78)(25,147,202)(26,148,203)(27,149,204)(28,150,205)(29,151,206)(30,152,207)(31,145,208)(32,146,201)(33,214,159)(34,215,160)(35,216,153)(36,209,154)(37,210,155)(38,211,156)(39,212,157)(40,213,158)(41,233,89)(42,234,90)(43,235,91)(44,236,92)(45,237,93)(46,238,94)(47,239,95)(48,240,96)(49,194,106)(50,195,107)(51,196,108)(52,197,109)(53,198,110)(54,199,111)(55,200,112)(56,193,105)(97,175,123)(98,176,124)(99,169,125)(100,170,126)(101,171,127)(102,172,128)(103,173,121)(104,174,122)(129,144,161)(130,137,162)(131,138,163)(132,139,164)(133,140,165)(134,141,166)(135,142,167)(136,143,168)(177,232,218)(178,225,219)(179,226,220)(180,227,221)(181,228,222)(182,229,223)(183,230,224)(184,231,217), (1,172)(2,173)(3,174)(4,175)(5,176)(6,169)(7,170)(8,171)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,105)(17,149)(18,150)(19,151)(20,152)(21,145)(22,146)(23,147)(24,148)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,129)(41,226)(42,227)(43,228)(44,229)(45,230)(46,231)(47,232)(48,225)(49,188)(50,189)(51,190)(52,191)(53,192)(54,185)(55,186)(56,187)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,206)(74,207)(75,208)(76,201)(77,202)(78,203)(79,204)(80,205)(81,195)(82,196)(83,197)(84,198)(85,199)(86,200)(87,193)(88,194)(89,220)(90,221)(91,222)(92,223)(93,224)(94,217)(95,218)(96,219)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,121)(137,159)(138,160)(139,153)(140,154)(141,155)(142,156)(143,157)(144,158)(161,213)(162,214)(163,215)(164,216)(165,209)(166,210)(167,211)(168,212)(177,239)(178,240)(179,233)(180,234)(181,235)(182,236)(183,237)(184,238), (1,78,194,183,40)(2,33,184,195,79)(3,80,196,177,34)(4,35,178,197,73)(5,74,198,179,36)(6,37,180,199,75)(7,76,200,181,38)(8,39,182,193,77)(9,45,161,102,148)(10,149,103,162,46)(11,47,163,104,150)(12,151,97,164,48)(13,41,165,98,152)(14,145,99,166,42)(15,43,167,100,146)(16,147,101,168,44)(17,69,214,231,107)(18,108,232,215,70)(19,71,216,225,109)(20,110,226,209,72)(21,65,210,227,111)(22,112,228,211,66)(23,67,212,229,105)(24,106,230,213,68)(25,127,143,92,187)(26,188,93,144,128)(27,121,137,94,189)(28,190,95,138,122)(29,123,139,96,191)(30,192,89,140,124)(31,125,141,90,185)(32,186,91,142,126)(49,224,158,119,61)(50,62,120,159,217)(51,218,160,113,63)(52,64,114,153,219)(53,220,154,115,57)(54,58,116,155,221)(55,222,156,117,59)(56,60,118,157,223)(81,204,173,130,238)(82,239,131,174,205)(83,206,175,132,240)(84,233,133,176,207)(85,208,169,134,234)(86,235,135,170,201)(87,202,171,136,236)(88,237,129,172,203), (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)>;
G:=Group( (1,176)(2,169)(3,170)(4,171)(5,172)(6,173)(7,174)(8,175)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,134)(34,135)(35,136)(36,129)(37,130)(38,131)(39,132)(40,133)(41,224)(42,217)(43,218)(44,219)(45,220)(46,221)(47,222)(48,223)(57,148)(58,149)(59,150)(60,151)(61,152)(62,145)(63,146)(64,147)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,202)(74,203)(75,204)(76,205)(77,206)(78,207)(79,208)(80,201)(81,199)(82,200)(83,193)(84,194)(85,195)(86,196)(87,197)(88,198)(89,230)(90,231)(91,232)(92,225)(93,226)(94,227)(95,228)(96,229)(97,118)(98,119)(99,120)(100,113)(101,114)(102,115)(103,116)(104,117)(105,191)(106,192)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(137,210)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,209)(153,168)(154,161)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(177,235)(178,236)(179,237)(180,238)(181,239)(182,240)(183,233)(184,234), (1,68,119)(2,69,120)(3,70,113)(4,71,114)(5,72,115)(6,65,116)(7,66,117)(8,67,118)(9,88,188)(10,81,189)(11,82,190)(12,83,191)(13,84,192)(14,85,185)(15,86,186)(16,87,187)(17,62,79)(18,63,80)(19,64,73)(20,57,74)(21,58,75)(22,59,76)(23,60,77)(24,61,78)(25,147,202)(26,148,203)(27,149,204)(28,150,205)(29,151,206)(30,152,207)(31,145,208)(32,146,201)(33,214,159)(34,215,160)(35,216,153)(36,209,154)(37,210,155)(38,211,156)(39,212,157)(40,213,158)(41,233,89)(42,234,90)(43,235,91)(44,236,92)(45,237,93)(46,238,94)(47,239,95)(48,240,96)(49,194,106)(50,195,107)(51,196,108)(52,197,109)(53,198,110)(54,199,111)(55,200,112)(56,193,105)(97,175,123)(98,176,124)(99,169,125)(100,170,126)(101,171,127)(102,172,128)(103,173,121)(104,174,122)(129,144,161)(130,137,162)(131,138,163)(132,139,164)(133,140,165)(134,141,166)(135,142,167)(136,143,168)(177,232,218)(178,225,219)(179,226,220)(180,227,221)(181,228,222)(182,229,223)(183,230,224)(184,231,217), (1,172)(2,173)(3,174)(4,175)(5,176)(6,169)(7,170)(8,171)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,105)(17,149)(18,150)(19,151)(20,152)(21,145)(22,146)(23,147)(24,148)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,129)(41,226)(42,227)(43,228)(44,229)(45,230)(46,231)(47,232)(48,225)(49,188)(50,189)(51,190)(52,191)(53,192)(54,185)(55,186)(56,187)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,206)(74,207)(75,208)(76,201)(77,202)(78,203)(79,204)(80,205)(81,195)(82,196)(83,197)(84,198)(85,199)(86,200)(87,193)(88,194)(89,220)(90,221)(91,222)(92,223)(93,224)(94,217)(95,218)(96,219)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,121)(137,159)(138,160)(139,153)(140,154)(141,155)(142,156)(143,157)(144,158)(161,213)(162,214)(163,215)(164,216)(165,209)(166,210)(167,211)(168,212)(177,239)(178,240)(179,233)(180,234)(181,235)(182,236)(183,237)(184,238), (1,78,194,183,40)(2,33,184,195,79)(3,80,196,177,34)(4,35,178,197,73)(5,74,198,179,36)(6,37,180,199,75)(7,76,200,181,38)(8,39,182,193,77)(9,45,161,102,148)(10,149,103,162,46)(11,47,163,104,150)(12,151,97,164,48)(13,41,165,98,152)(14,145,99,166,42)(15,43,167,100,146)(16,147,101,168,44)(17,69,214,231,107)(18,108,232,215,70)(19,71,216,225,109)(20,110,226,209,72)(21,65,210,227,111)(22,112,228,211,66)(23,67,212,229,105)(24,106,230,213,68)(25,127,143,92,187)(26,188,93,144,128)(27,121,137,94,189)(28,190,95,138,122)(29,123,139,96,191)(30,192,89,140,124)(31,125,141,90,185)(32,186,91,142,126)(49,224,158,119,61)(50,62,120,159,217)(51,218,160,113,63)(52,64,114,153,219)(53,220,154,115,57)(54,58,116,155,221)(55,222,156,117,59)(56,60,118,157,223)(81,204,173,130,238)(82,239,131,174,205)(83,206,175,132,240)(84,233,133,176,207)(85,208,169,134,234)(86,235,135,170,201)(87,202,171,136,236)(88,237,129,172,203), (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) );
G=PermutationGroup([[(1,176),(2,169),(3,170),(4,171),(5,172),(6,173),(7,174),(8,175),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,134),(34,135),(35,136),(36,129),(37,130),(38,131),(39,132),(40,133),(41,224),(42,217),(43,218),(44,219),(45,220),(46,221),(47,222),(48,223),(57,148),(58,149),(59,150),(60,151),(61,152),(62,145),(63,146),(64,147),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,127),(72,128),(73,202),(74,203),(75,204),(76,205),(77,206),(78,207),(79,208),(80,201),(81,199),(82,200),(83,193),(84,194),(85,195),(86,196),(87,197),(88,198),(89,230),(90,231),(91,232),(92,225),(93,226),(94,227),(95,228),(96,229),(97,118),(98,119),(99,120),(100,113),(101,114),(102,115),(103,116),(104,117),(105,191),(106,192),(107,185),(108,186),(109,187),(110,188),(111,189),(112,190),(137,210),(138,211),(139,212),(140,213),(141,214),(142,215),(143,216),(144,209),(153,168),(154,161),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(177,235),(178,236),(179,237),(180,238),(181,239),(182,240),(183,233),(184,234)], [(1,68,119),(2,69,120),(3,70,113),(4,71,114),(5,72,115),(6,65,116),(7,66,117),(8,67,118),(9,88,188),(10,81,189),(11,82,190),(12,83,191),(13,84,192),(14,85,185),(15,86,186),(16,87,187),(17,62,79),(18,63,80),(19,64,73),(20,57,74),(21,58,75),(22,59,76),(23,60,77),(24,61,78),(25,147,202),(26,148,203),(27,149,204),(28,150,205),(29,151,206),(30,152,207),(31,145,208),(32,146,201),(33,214,159),(34,215,160),(35,216,153),(36,209,154),(37,210,155),(38,211,156),(39,212,157),(40,213,158),(41,233,89),(42,234,90),(43,235,91),(44,236,92),(45,237,93),(46,238,94),(47,239,95),(48,240,96),(49,194,106),(50,195,107),(51,196,108),(52,197,109),(53,198,110),(54,199,111),(55,200,112),(56,193,105),(97,175,123),(98,176,124),(99,169,125),(100,170,126),(101,171,127),(102,172,128),(103,173,121),(104,174,122),(129,144,161),(130,137,162),(131,138,163),(132,139,164),(133,140,165),(134,141,166),(135,142,167),(136,143,168),(177,232,218),(178,225,219),(179,226,220),(180,227,221),(181,228,222),(182,229,223),(183,230,224),(184,231,217)], [(1,172),(2,173),(3,174),(4,175),(5,176),(6,169),(7,170),(8,171),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,105),(17,149),(18,150),(19,151),(20,152),(21,145),(22,146),(23,147),(24,148),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,129),(41,226),(42,227),(43,228),(44,229),(45,230),(46,231),(47,232),(48,225),(49,188),(50,189),(51,190),(52,191),(53,192),(54,185),(55,186),(56,187),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,97),(72,98),(73,206),(74,207),(75,208),(76,201),(77,202),(78,203),(79,204),(80,205),(81,195),(82,196),(83,197),(84,198),(85,199),(86,200),(87,193),(88,194),(89,220),(90,221),(91,222),(92,223),(93,224),(94,217),(95,218),(96,219),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,121),(137,159),(138,160),(139,153),(140,154),(141,155),(142,156),(143,157),(144,158),(161,213),(162,214),(163,215),(164,216),(165,209),(166,210),(167,211),(168,212),(177,239),(178,240),(179,233),(180,234),(181,235),(182,236),(183,237),(184,238)], [(1,78,194,183,40),(2,33,184,195,79),(3,80,196,177,34),(4,35,178,197,73),(5,74,198,179,36),(6,37,180,199,75),(7,76,200,181,38),(8,39,182,193,77),(9,45,161,102,148),(10,149,103,162,46),(11,47,163,104,150),(12,151,97,164,48),(13,41,165,98,152),(14,145,99,166,42),(15,43,167,100,146),(16,147,101,168,44),(17,69,214,231,107),(18,108,232,215,70),(19,71,216,225,109),(20,110,226,209,72),(21,65,210,227,111),(22,112,228,211,66),(23,67,212,229,105),(24,106,230,213,68),(25,127,143,92,187),(26,188,93,144,128),(27,121,137,94,189),(28,190,95,138,122),(29,123,139,96,191),(30,192,89,140,124),(31,125,141,90,185),(32,186,91,142,126),(49,224,158,119,61),(50,62,120,159,217),(51,218,160,113,63),(52,64,114,153,219),(53,220,154,115,57),(54,58,116,155,221),(55,222,156,117,59),(56,60,118,157,223),(81,204,173,130,238),(82,239,131,174,205),(83,206,175,132,240),(84,233,133,176,207),(85,208,169,134,234),(86,235,135,170,201),(87,202,171,136,236),(88,237,129,172,203)], [(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)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 15 | ··· | 15 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | + | - | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D5 | D6 | D6 | Dic5 | D10 | Dic5 | D10 | Dic5 | C4×S3 | C4×S3 | C5⋊2C8 | S3×C8 | S3×D5 | S3×Dic5 | C2×S3×D5 | S3×Dic5 | S3×C5⋊2C8 |
| kernel | C2×S3×C5⋊2C8 | S3×C5⋊2C8 | C6×C5⋊2C8 | C2×C15⋊3C8 | S3×C2×C20 | S3×C20 | C10×Dic3 | S3×C2×C10 | S3×C10 | C2×C5⋊2C8 | S3×C2×C4 | C5⋊2C8 | C2×C20 | C4×S3 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C2×C10 | D6 | C10 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 16 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 16 | 8 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×S3×C5⋊2C8 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 144 |
| 0 | 0 | 5 | 239 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 97 |
| 0 | 0 | 0 | 1 |
| 189 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 176 | 142 | 0 | 0 |
| 148 | 65 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,1,5,0,0,144,239],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,97,1],[189,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[176,148,0,0,142,65,0,0,0,0,64,0,0,0,0,64] >;
C2×S3×C5⋊2C8 in GAP, Magma, Sage, TeX
C_2\times S_3\times C_5\rtimes_2C_8
% in TeX
G:=Group("C2xS3xC5:2C8"); // GroupNames label
G:=SmallGroup(480,361);
// by ID
G=gap.SmallGroup(480,361);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^5=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations