direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×D5×C3⋊C8, C60.174C23, C6⋊3(C8×D5), C30⋊5(C2×C8), (C6×D5)⋊2C8, C15⋊7(C22×C8), (D5×C12).4C4, C12.78(C4×D5), C60.147(C2×C4), (C4×D5).100D6, (C2×C20).327D6, (C6×Dic5).9C4, C4.22(D5×Dic3), C15⋊3C8⋊42C22, (C2×C12).331D10, C30.96(C22×C4), (C4×D5).10Dic3, C20.48(C2×Dic3), (C2×C60).229C22, C20.171(C22×S3), D10.18(C2×Dic3), C12.171(C22×D5), C22.13(D5×Dic3), (C2×Dic5).16Dic3, Dic5.20(C2×Dic3), (D5×C12).101C22, (C22×D5).10Dic3, C10.14(C22×Dic3), C3⋊4(D5×C2×C8), C10⋊3(C2×C3⋊C8), C5⋊3(C22×C3⋊C8), (C10×C3⋊C8)⋊11C2, (D5×C2×C6).6C4, C6.77(C2×C4×D5), (C3×D5)⋊3(C2×C8), (C2×C4×D5).20S3, (D5×C2×C12).8C2, C4.144(C2×S3×D5), C2.1(C2×D5×Dic3), (C5×C3⋊C8)⋊34C22, (C2×C6).50(C4×D5), (C2×C15⋊3C8)⋊23C2, (C2×C30).93(C2×C4), (C6×D5).47(C2×C4), (C2×C4).232(S3×D5), (C3×Dic5).55(C2×C4), (C2×C10).34(C2×Dic3), SmallGroup(480,357)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C2×D5×C3⋊C8 |
Generators and relations for C2×D5×C3⋊C8
G = < a,b,c,d,e | a2=b5=c2=d3=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: 476 in 152 conjugacy classes, 84 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C3⋊C8, C22×C12, C3×Dic5, C60, C6×D5, C2×C30, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C22×C3⋊C8, C5×C3⋊C8, C15⋊3C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×C2×C8, D5×C3⋊C8, C10×C3⋊C8, C2×C15⋊3C8, D5×C2×C12, C2×D5×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D5, Dic3, D6, C2×C8, C22×C4, D10, C3⋊C8, C2×Dic3, C22×S3, C22×C8, C4×D5, C22×D5, C2×C3⋊C8, C22×Dic3, S3×D5, C8×D5, C2×C4×D5, C22×C3⋊C8, D5×Dic3, C2×S3×D5, D5×C2×C8, D5×C3⋊C8, C2×D5×Dic3, C2×D5×C3⋊C8
►On 240 points
Generators in S240
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(25 117)(26 118)(27 119)(28 120)(29 113)(30 114)(31 115)(32 116)(33 94)(34 95)(35 96)(36 89)(37 90)(38 91)(39 92)(40 93)(41 133)(42 134)(43 135)(44 136)(45 129)(46 130)(47 131)(48 132)(49 127)(50 128)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 112)(58 105)(59 106)(60 107)(61 108)(62 109)(63 110)(64 111)(73 221)(74 222)(75 223)(76 224)(77 217)(78 218)(79 219)(80 220)(81 215)(82 216)(83 209)(84 210)(85 211)(86 212)(87 213)(88 214)(97 237)(98 238)(99 239)(100 240)(101 233)(102 234)(103 235)(104 236)(137 187)(138 188)(139 189)(140 190)(141 191)(142 192)(143 185)(144 186)(145 181)(146 182)(147 183)(148 184)(149 177)(150 178)(151 179)(152 180)(153 203)(154 204)(155 205)(156 206)(157 207)(158 208)(159 201)(160 202)(161 197)(162 198)(163 199)(164 200)(165 193)(166 194)(167 195)(168 196)(169 228)(170 229)(171 230)(172 231)(173 232)(174 225)(175 226)(176 227)
(1 193 233 121 155)(2 194 234 122 156)(3 195 235 123 157)(4 196 236 124 158)(5 197 237 125 159)(6 198 238 126 160)(7 199 239 127 153)(8 200 240 128 154)(9 216 110 229 45)(10 209 111 230 46)(11 210 112 231 47)(12 211 105 232 48)(13 212 106 225 41)(14 213 107 226 42)(15 214 108 227 43)(16 215 109 228 44)(17 163 99 49 203)(18 164 100 50 204)(19 165 101 51 205)(20 166 102 52 206)(21 167 103 53 207)(22 168 104 54 208)(23 161 97 55 201)(24 162 98 56 202)(25 179 143 77 94)(26 180 144 78 95)(27 181 137 79 96)(28 182 138 80 89)(29 183 139 73 90)(30 184 140 74 91)(31 177 141 75 92)(32 178 142 76 93)(33 117 151 185 217)(34 118 152 186 218)(35 119 145 187 219)(36 120 146 188 220)(37 113 147 189 221)(38 114 148 190 222)(39 115 149 191 223)(40 116 150 192 224)(57 172 131 65 84)(58 173 132 66 85)(59 174 133 67 86)(60 175 134 68 87)(61 176 135 69 88)(62 169 136 70 81)(63 170 129 71 82)(64 171 130 72 83)
(1 159)(2 160)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 225)(10 226)(11 227)(12 228)(13 229)(14 230)(15 231)(16 232)(17 207)(18 208)(19 201)(20 202)(21 203)(22 204)(23 205)(24 206)(25 183)(26 184)(27 177)(28 178)(29 179)(30 180)(31 181)(32 182)(33 189)(34 190)(35 191)(36 192)(37 185)(38 186)(39 187)(40 188)(41 45)(42 46)(43 47)(44 48)(49 167)(50 168)(51 161)(52 162)(53 163)(54 164)(55 165)(56 166)(57 88)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 176)(66 169)(67 170)(68 171)(69 172)(70 173)(71 174)(72 175)(73 77)(74 78)(75 79)(76 80)(89 142)(90 143)(91 144)(92 137)(93 138)(94 139)(95 140)(96 141)(97 101)(98 102)(99 103)(100 104)(105 215)(106 216)(107 209)(108 210)(109 211)(110 212)(111 213)(112 214)(113 151)(114 152)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)(121 197)(122 198)(123 199)(124 200)(125 193)(126 194)(127 195)(128 196)(129 133)(130 134)(131 135)(132 136)(217 221)(218 222)(219 223)(220 224)(233 237)(234 238)(235 239)(236 240)
(1 184 106)(2 107 177)(3 178 108)(4 109 179)(5 180 110)(6 111 181)(7 182 112)(8 105 183)(9 125 95)(10 96 126)(11 127 89)(12 90 128)(13 121 91)(14 92 122)(15 123 93)(16 94 124)(17 146 57)(18 58 147)(19 148 59)(20 60 149)(21 150 61)(22 62 151)(23 152 63)(24 64 145)(25 158 215)(26 216 159)(27 160 209)(28 210 153)(29 154 211)(30 212 155)(31 156 213)(32 214 157)(33 54 70)(34 71 55)(35 56 72)(36 65 49)(37 50 66)(38 67 51)(39 52 68)(40 69 53)(41 233 74)(42 75 234)(43 235 76)(44 77 236)(45 237 78)(46 79 238)(47 239 80)(48 73 240)(81 117 208)(82 201 118)(83 119 202)(84 203 120)(85 113 204)(86 205 114)(87 115 206)(88 207 116)(97 218 129)(98 130 219)(99 220 131)(100 132 221)(101 222 133)(102 134 223)(103 224 135)(104 136 217)(137 198 230)(138 231 199)(139 200 232)(140 225 193)(141 194 226)(142 227 195)(143 196 228)(144 229 197)(161 186 170)(162 171 187)(163 188 172)(164 173 189)(165 190 174)(166 175 191)(167 192 176)(168 169 185)
(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)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(25,117)(26,118)(27,119)(28,120)(29,113)(30,114)(31,115)(32,116)(33,94)(34,95)(35,96)(36,89)(37,90)(38,91)(39,92)(40,93)(41,133)(42,134)(43,135)(44,136)(45,129)(46,130)(47,131)(48,132)(49,127)(50,128)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,112)(58,105)(59,106)(60,107)(61,108)(62,109)(63,110)(64,111)(73,221)(74,222)(75,223)(76,224)(77,217)(78,218)(79,219)(80,220)(81,215)(82,216)(83,209)(84,210)(85,211)(86,212)(87,213)(88,214)(97,237)(98,238)(99,239)(100,240)(101,233)(102,234)(103,235)(104,236)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,185)(144,186)(145,181)(146,182)(147,183)(148,184)(149,177)(150,178)(151,179)(152,180)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,201)(160,202)(161,197)(162,198)(163,199)(164,200)(165,193)(166,194)(167,195)(168,196)(169,228)(170,229)(171,230)(172,231)(173,232)(174,225)(175,226)(176,227), (1,193,233,121,155)(2,194,234,122,156)(3,195,235,123,157)(4,196,236,124,158)(5,197,237,125,159)(6,198,238,126,160)(7,199,239,127,153)(8,200,240,128,154)(9,216,110,229,45)(10,209,111,230,46)(11,210,112,231,47)(12,211,105,232,48)(13,212,106,225,41)(14,213,107,226,42)(15,214,108,227,43)(16,215,109,228,44)(17,163,99,49,203)(18,164,100,50,204)(19,165,101,51,205)(20,166,102,52,206)(21,167,103,53,207)(22,168,104,54,208)(23,161,97,55,201)(24,162,98,56,202)(25,179,143,77,94)(26,180,144,78,95)(27,181,137,79,96)(28,182,138,80,89)(29,183,139,73,90)(30,184,140,74,91)(31,177,141,75,92)(32,178,142,76,93)(33,117,151,185,217)(34,118,152,186,218)(35,119,145,187,219)(36,120,146,188,220)(37,113,147,189,221)(38,114,148,190,222)(39,115,149,191,223)(40,116,150,192,224)(57,172,131,65,84)(58,173,132,66,85)(59,174,133,67,86)(60,175,134,68,87)(61,176,135,69,88)(62,169,136,70,81)(63,170,129,71,82)(64,171,130,72,83), (1,159)(2,160)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,225)(10,226)(11,227)(12,228)(13,229)(14,230)(15,231)(16,232)(17,207)(18,208)(19,201)(20,202)(21,203)(22,204)(23,205)(24,206)(25,183)(26,184)(27,177)(28,178)(29,179)(30,180)(31,181)(32,182)(33,189)(34,190)(35,191)(36,192)(37,185)(38,186)(39,187)(40,188)(41,45)(42,46)(43,47)(44,48)(49,167)(50,168)(51,161)(52,162)(53,163)(54,164)(55,165)(56,166)(57,88)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,176)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,77)(74,78)(75,79)(76,80)(89,142)(90,143)(91,144)(92,137)(93,138)(94,139)(95,140)(96,141)(97,101)(98,102)(99,103)(100,104)(105,215)(106,216)(107,209)(108,210)(109,211)(110,212)(111,213)(112,214)(113,151)(114,152)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)(121,197)(122,198)(123,199)(124,200)(125,193)(126,194)(127,195)(128,196)(129,133)(130,134)(131,135)(132,136)(217,221)(218,222)(219,223)(220,224)(233,237)(234,238)(235,239)(236,240), (1,184,106)(2,107,177)(3,178,108)(4,109,179)(5,180,110)(6,111,181)(7,182,112)(8,105,183)(9,125,95)(10,96,126)(11,127,89)(12,90,128)(13,121,91)(14,92,122)(15,123,93)(16,94,124)(17,146,57)(18,58,147)(19,148,59)(20,60,149)(21,150,61)(22,62,151)(23,152,63)(24,64,145)(25,158,215)(26,216,159)(27,160,209)(28,210,153)(29,154,211)(30,212,155)(31,156,213)(32,214,157)(33,54,70)(34,71,55)(35,56,72)(36,65,49)(37,50,66)(38,67,51)(39,52,68)(40,69,53)(41,233,74)(42,75,234)(43,235,76)(44,77,236)(45,237,78)(46,79,238)(47,239,80)(48,73,240)(81,117,208)(82,201,118)(83,119,202)(84,203,120)(85,113,204)(86,205,114)(87,115,206)(88,207,116)(97,218,129)(98,130,219)(99,220,131)(100,132,221)(101,222,133)(102,134,223)(103,224,135)(104,136,217)(137,198,230)(138,231,199)(139,200,232)(140,225,193)(141,194,226)(142,227,195)(143,196,228)(144,229,197)(161,186,170)(162,171,187)(163,188,172)(164,173,189)(165,190,174)(166,175,191)(167,192,176)(168,169,185), (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( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(25,117)(26,118)(27,119)(28,120)(29,113)(30,114)(31,115)(32,116)(33,94)(34,95)(35,96)(36,89)(37,90)(38,91)(39,92)(40,93)(41,133)(42,134)(43,135)(44,136)(45,129)(46,130)(47,131)(48,132)(49,127)(50,128)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,112)(58,105)(59,106)(60,107)(61,108)(62,109)(63,110)(64,111)(73,221)(74,222)(75,223)(76,224)(77,217)(78,218)(79,219)(80,220)(81,215)(82,216)(83,209)(84,210)(85,211)(86,212)(87,213)(88,214)(97,237)(98,238)(99,239)(100,240)(101,233)(102,234)(103,235)(104,236)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,185)(144,186)(145,181)(146,182)(147,183)(148,184)(149,177)(150,178)(151,179)(152,180)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,201)(160,202)(161,197)(162,198)(163,199)(164,200)(165,193)(166,194)(167,195)(168,196)(169,228)(170,229)(171,230)(172,231)(173,232)(174,225)(175,226)(176,227), (1,193,233,121,155)(2,194,234,122,156)(3,195,235,123,157)(4,196,236,124,158)(5,197,237,125,159)(6,198,238,126,160)(7,199,239,127,153)(8,200,240,128,154)(9,216,110,229,45)(10,209,111,230,46)(11,210,112,231,47)(12,211,105,232,48)(13,212,106,225,41)(14,213,107,226,42)(15,214,108,227,43)(16,215,109,228,44)(17,163,99,49,203)(18,164,100,50,204)(19,165,101,51,205)(20,166,102,52,206)(21,167,103,53,207)(22,168,104,54,208)(23,161,97,55,201)(24,162,98,56,202)(25,179,143,77,94)(26,180,144,78,95)(27,181,137,79,96)(28,182,138,80,89)(29,183,139,73,90)(30,184,140,74,91)(31,177,141,75,92)(32,178,142,76,93)(33,117,151,185,217)(34,118,152,186,218)(35,119,145,187,219)(36,120,146,188,220)(37,113,147,189,221)(38,114,148,190,222)(39,115,149,191,223)(40,116,150,192,224)(57,172,131,65,84)(58,173,132,66,85)(59,174,133,67,86)(60,175,134,68,87)(61,176,135,69,88)(62,169,136,70,81)(63,170,129,71,82)(64,171,130,72,83), (1,159)(2,160)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,225)(10,226)(11,227)(12,228)(13,229)(14,230)(15,231)(16,232)(17,207)(18,208)(19,201)(20,202)(21,203)(22,204)(23,205)(24,206)(25,183)(26,184)(27,177)(28,178)(29,179)(30,180)(31,181)(32,182)(33,189)(34,190)(35,191)(36,192)(37,185)(38,186)(39,187)(40,188)(41,45)(42,46)(43,47)(44,48)(49,167)(50,168)(51,161)(52,162)(53,163)(54,164)(55,165)(56,166)(57,88)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,176)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,77)(74,78)(75,79)(76,80)(89,142)(90,143)(91,144)(92,137)(93,138)(94,139)(95,140)(96,141)(97,101)(98,102)(99,103)(100,104)(105,215)(106,216)(107,209)(108,210)(109,211)(110,212)(111,213)(112,214)(113,151)(114,152)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)(121,197)(122,198)(123,199)(124,200)(125,193)(126,194)(127,195)(128,196)(129,133)(130,134)(131,135)(132,136)(217,221)(218,222)(219,223)(220,224)(233,237)(234,238)(235,239)(236,240), (1,184,106)(2,107,177)(3,178,108)(4,109,179)(5,180,110)(6,111,181)(7,182,112)(8,105,183)(9,125,95)(10,96,126)(11,127,89)(12,90,128)(13,121,91)(14,92,122)(15,123,93)(16,94,124)(17,146,57)(18,58,147)(19,148,59)(20,60,149)(21,150,61)(22,62,151)(23,152,63)(24,64,145)(25,158,215)(26,216,159)(27,160,209)(28,210,153)(29,154,211)(30,212,155)(31,156,213)(32,214,157)(33,54,70)(34,71,55)(35,56,72)(36,65,49)(37,50,66)(38,67,51)(39,52,68)(40,69,53)(41,233,74)(42,75,234)(43,235,76)(44,77,236)(45,237,78)(46,79,238)(47,239,80)(48,73,240)(81,117,208)(82,201,118)(83,119,202)(84,203,120)(85,113,204)(86,205,114)(87,115,206)(88,207,116)(97,218,129)(98,130,219)(99,220,131)(100,132,221)(101,222,133)(102,134,223)(103,224,135)(104,136,217)(137,198,230)(138,231,199)(139,200,232)(140,225,193)(141,194,226)(142,227,195)(143,196,228)(144,229,197)(161,186,170)(162,171,187)(163,188,172)(164,173,189)(165,190,174)(166,175,191)(167,192,176)(168,169,185), (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([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,71),(10,72),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(25,117),(26,118),(27,119),(28,120),(29,113),(30,114),(31,115),(32,116),(33,94),(34,95),(35,96),(36,89),(37,90),(38,91),(39,92),(40,93),(41,133),(42,134),(43,135),(44,136),(45,129),(46,130),(47,131),(48,132),(49,127),(50,128),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,112),(58,105),(59,106),(60,107),(61,108),(62,109),(63,110),(64,111),(73,221),(74,222),(75,223),(76,224),(77,217),(78,218),(79,219),(80,220),(81,215),(82,216),(83,209),(84,210),(85,211),(86,212),(87,213),(88,214),(97,237),(98,238),(99,239),(100,240),(101,233),(102,234),(103,235),(104,236),(137,187),(138,188),(139,189),(140,190),(141,191),(142,192),(143,185),(144,186),(145,181),(146,182),(147,183),(148,184),(149,177),(150,178),(151,179),(152,180),(153,203),(154,204),(155,205),(156,206),(157,207),(158,208),(159,201),(160,202),(161,197),(162,198),(163,199),(164,200),(165,193),(166,194),(167,195),(168,196),(169,228),(170,229),(171,230),(172,231),(173,232),(174,225),(175,226),(176,227)], [(1,193,233,121,155),(2,194,234,122,156),(3,195,235,123,157),(4,196,236,124,158),(5,197,237,125,159),(6,198,238,126,160),(7,199,239,127,153),(8,200,240,128,154),(9,216,110,229,45),(10,209,111,230,46),(11,210,112,231,47),(12,211,105,232,48),(13,212,106,225,41),(14,213,107,226,42),(15,214,108,227,43),(16,215,109,228,44),(17,163,99,49,203),(18,164,100,50,204),(19,165,101,51,205),(20,166,102,52,206),(21,167,103,53,207),(22,168,104,54,208),(23,161,97,55,201),(24,162,98,56,202),(25,179,143,77,94),(26,180,144,78,95),(27,181,137,79,96),(28,182,138,80,89),(29,183,139,73,90),(30,184,140,74,91),(31,177,141,75,92),(32,178,142,76,93),(33,117,151,185,217),(34,118,152,186,218),(35,119,145,187,219),(36,120,146,188,220),(37,113,147,189,221),(38,114,148,190,222),(39,115,149,191,223),(40,116,150,192,224),(57,172,131,65,84),(58,173,132,66,85),(59,174,133,67,86),(60,175,134,68,87),(61,176,135,69,88),(62,169,136,70,81),(63,170,129,71,82),(64,171,130,72,83)], [(1,159),(2,160),(3,153),(4,154),(5,155),(6,156),(7,157),(8,158),(9,225),(10,226),(11,227),(12,228),(13,229),(14,230),(15,231),(16,232),(17,207),(18,208),(19,201),(20,202),(21,203),(22,204),(23,205),(24,206),(25,183),(26,184),(27,177),(28,178),(29,179),(30,180),(31,181),(32,182),(33,189),(34,190),(35,191),(36,192),(37,185),(38,186),(39,187),(40,188),(41,45),(42,46),(43,47),(44,48),(49,167),(50,168),(51,161),(52,162),(53,163),(54,164),(55,165),(56,166),(57,88),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,176),(66,169),(67,170),(68,171),(69,172),(70,173),(71,174),(72,175),(73,77),(74,78),(75,79),(76,80),(89,142),(90,143),(91,144),(92,137),(93,138),(94,139),(95,140),(96,141),(97,101),(98,102),(99,103),(100,104),(105,215),(106,216),(107,209),(108,210),(109,211),(110,212),(111,213),(112,214),(113,151),(114,152),(115,145),(116,146),(117,147),(118,148),(119,149),(120,150),(121,197),(122,198),(123,199),(124,200),(125,193),(126,194),(127,195),(128,196),(129,133),(130,134),(131,135),(132,136),(217,221),(218,222),(219,223),(220,224),(233,237),(234,238),(235,239),(236,240)], [(1,184,106),(2,107,177),(3,178,108),(4,109,179),(5,180,110),(6,111,181),(7,182,112),(8,105,183),(9,125,95),(10,96,126),(11,127,89),(12,90,128),(13,121,91),(14,92,122),(15,123,93),(16,94,124),(17,146,57),(18,58,147),(19,148,59),(20,60,149),(21,150,61),(22,62,151),(23,152,63),(24,64,145),(25,158,215),(26,216,159),(27,160,209),(28,210,153),(29,154,211),(30,212,155),(31,156,213),(32,214,157),(33,54,70),(34,71,55),(35,56,72),(36,65,49),(37,50,66),(38,67,51),(39,52,68),(40,69,53),(41,233,74),(42,75,234),(43,235,76),(44,77,236),(45,237,78),(46,79,238),(47,239,80),(48,73,240),(81,117,208),(82,201,118),(83,119,202),(84,203,120),(85,113,204),(86,205,114),(87,115,206),(88,207,116),(97,218,129),(98,130,219),(99,220,131),(100,132,221),(101,222,133),(102,134,223),(103,224,135),(104,136,217),(137,198,230),(138,231,199),(139,200,232),(140,225,193),(141,194,226),(142,227,195),(143,196,228),(144,229,197),(161,186,170),(162,171,187),(163,188,172),(164,173,189),(165,190,174),(166,175,191),(167,192,176),(168,169,185)], [(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 | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 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 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 3 | ··· | 3 | 15 | ··· | 15 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 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 | Dic3 | D6 | Dic3 | D6 | Dic3 | D10 | D10 | C3⋊C8 | C4×D5 | C4×D5 | C8×D5 | S3×D5 | D5×Dic3 | C2×S3×D5 | D5×Dic3 | D5×C3⋊C8 |
| kernel | C2×D5×C3⋊C8 | D5×C3⋊C8 | C10×C3⋊C8 | C2×C15⋊3C8 | D5×C2×C12 | D5×C12 | C6×Dic5 | D5×C2×C6 | C6×D5 | C2×C4×D5 | C2×C3⋊C8 | C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C3⋊C8 | C2×C12 | D10 | C12 | C2×C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 16 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×D5×C3⋊C8 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 1 | 0 | 0 |
| 188 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 53 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 240 |
| 211 | 0 | 0 | 0 |
| 0 | 211 | 0 | 0 |
| 0 | 0 | 170 | 16 |
| 0 | 0 | 186 | 71 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,188,0,0,1,52,0,0,0,0,1,0,0,0,0,1],[1,53,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,240],[211,0,0,0,0,211,0,0,0,0,170,186,0,0,16,71] >;
C2×D5×C3⋊C8 in GAP, Magma, Sage, TeX
C_2\times D_5\times C_3\rtimes C_8
% in TeX
G:=Group("C2xD5xC3:C8"); // GroupNames label
G:=SmallGroup(480,357);
// by ID
G=gap.SmallGroup(480,357);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^3=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