direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8.11D6, C60.152D4, C60.231C23, (C6×Q8)⋊2C10, (Q8×C10)⋊13S3, (Q8×C30)⋊16C2, C3⋊Q16⋊5C10, C6.54(D4×C10), C12.19(C5×D4), (C5×Q8).55D6, Q8⋊2S3⋊5C10, C4○D12.5C10, C30.437(C2×D4), (C2×C20).245D6, (C2×C30).185D4, C4.Dic3⋊7C10, Q8.11(S3×C10), D12.10(C2×C10), C20.96(C3⋊D4), C15⋊36(C8.C22), Dic6.9(C2×C10), C20.204(C22×S3), (C2×C60).365C22, C12.15(C22×C10), (C5×D12).49C22, (Q8×C15).49C22, (C5×Dic6).51C22, C3⋊C8.3(C2×C10), C4.15(S3×C2×C10), (C2×Q8)⋊4(C5×S3), C3⋊4(C5×C8.C22), (C2×C6).42(C5×D4), C4.17(C5×C3⋊D4), (C2×C4).18(S3×C10), (C5×C3⋊Q16)⋊13C2, C2.18(C10×C3⋊D4), (C5×C3⋊C8).29C22, (C3×Q8).6(C2×C10), (C2×C12).38(C2×C10), (C5×Q8⋊2S3)⋊13C2, (C5×C4○D12).11C2, C10.139(C2×C3⋊D4), (C5×C4.Dic3)⋊19C2, C22.11(C5×C3⋊D4), (C2×C10).64(C3⋊D4), SmallGroup(480,821)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8.11D6
G = < a,b,c,d,e | a5=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d5 >
Subgroups: 260 in 120 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C5×S3, C30, C30, C8.C22, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C4.Dic3, Q8⋊2S3, C3⋊Q16, C4○D12, C6×Q8, C5×Dic3, C60, C60, S3×C10, C2×C30, C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, Q8.11D6, C5×C3⋊C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C2×C60, Q8×C15, Q8×C15, C5×C8.C22, C5×C4.Dic3, C5×Q8⋊2S3, C5×C3⋊Q16, C5×C4○D12, Q8×C30, C5×Q8.11D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C8.C22, C5×D4, C22×C10, C2×C3⋊D4, S3×C10, D4×C10, Q8.11D6, C5×C3⋊D4, S3×C2×C10, C5×C8.C22, C10×C3⋊D4, C5×Q8.11D6
►On 240 points
Generators in S240
(1 56 42 30 22)(2 57 43 31 23)(3 58 44 32 24)(4 59 45 33 13)(5 60 46 34 14)(6 49 47 35 15)(7 50 48 36 16)(8 51 37 25 17)(9 52 38 26 18)(10 53 39 27 19)(11 54 40 28 20)(12 55 41 29 21)(61 111 108 90 83)(62 112 97 91 84)(63 113 98 92 73)(64 114 99 93 74)(65 115 100 94 75)(66 116 101 95 76)(67 117 102 96 77)(68 118 103 85 78)(69 119 104 86 79)(70 120 105 87 80)(71 109 106 88 81)(72 110 107 89 82)(121 175 157 154 136)(122 176 158 155 137)(123 177 159 156 138)(124 178 160 145 139)(125 179 161 146 140)(126 180 162 147 141)(127 169 163 148 142)(128 170 164 149 143)(129 171 165 150 144)(130 172 166 151 133)(131 173 167 152 134)(132 174 168 153 135)(181 235 217 205 199)(182 236 218 206 200)(183 237 219 207 201)(184 238 220 208 202)(185 239 221 209 203)(186 240 222 210 204)(187 229 223 211 193)(188 230 224 212 194)(189 231 225 213 195)(190 232 226 214 196)(191 233 227 215 197)(192 234 228 216 198)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 112 115 118)(110 113 116 119)(111 114 117 120)(121 130 127 124)(122 131 128 125)(123 132 129 126)(133 142 139 136)(134 143 140 137)(135 144 141 138)(145 154 151 148)(146 155 152 149)(147 156 153 150)(157 166 163 160)(158 167 164 161)(159 168 165 162)(169 178 175 172)(170 179 176 173)(171 180 177 174)(181 190 187 184)(182 191 188 185)(183 192 189 186)(193 202 199 196)(194 203 200 197)(195 204 201 198)(205 214 211 208)(206 215 212 209)(207 216 213 210)(217 226 223 220)(218 227 224 221)(219 228 225 222)(229 238 235 232)(230 239 236 233)(231 240 237 234)
(1 65 7 71)(2 72 8 66)(3 67 9 61)(4 62 10 68)(5 69 11 63)(6 64 12 70)(13 84 19 78)(14 79 20 73)(15 74 21 80)(16 81 22 75)(17 76 23 82)(18 83 24 77)(25 95 31 89)(26 90 32 96)(27 85 33 91)(28 92 34 86)(29 87 35 93)(30 94 36 88)(37 101 43 107)(38 108 44 102)(39 103 45 97)(40 98 46 104)(41 105 47 99)(42 100 48 106)(49 114 55 120)(50 109 56 115)(51 116 57 110)(52 111 58 117)(53 118 59 112)(54 113 60 119)(121 187 127 181)(122 182 128 188)(123 189 129 183)(124 184 130 190)(125 191 131 185)(126 186 132 192)(133 196 139 202)(134 203 140 197)(135 198 141 204)(136 193 142 199)(137 200 143 194)(138 195 144 201)(145 208 151 214)(146 215 152 209)(147 210 153 216)(148 205 154 211)(149 212 155 206)(150 207 156 213)(157 223 163 217)(158 218 164 224)(159 225 165 219)(160 220 166 226)(161 227 167 221)(162 222 168 228)(169 235 175 229)(170 230 176 236)(171 237 177 231)(172 232 178 238)(173 239 179 233)(174 234 180 240)
(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)
(1 123 7 129)(2 128 8 122)(3 121 9 127)(4 126 10 132)(5 131 11 125)(6 124 12 130)(13 141 19 135)(14 134 20 140)(15 139 21 133)(16 144 22 138)(17 137 23 143)(18 142 24 136)(25 155 31 149)(26 148 32 154)(27 153 33 147)(28 146 34 152)(29 151 35 145)(30 156 36 150)(37 158 43 164)(38 163 44 157)(39 168 45 162)(40 161 46 167)(41 166 47 160)(42 159 48 165)(49 178 55 172)(50 171 56 177)(51 176 57 170)(52 169 58 175)(53 174 59 180)(54 179 60 173)(61 184 67 190)(62 189 68 183)(63 182 69 188)(64 187 70 181)(65 192 71 186)(66 185 72 191)(73 200 79 194)(74 193 80 199)(75 198 81 204)(76 203 82 197)(77 196 83 202)(78 201 84 195)(85 207 91 213)(86 212 92 206)(87 205 93 211)(88 210 94 216)(89 215 95 209)(90 208 96 214)(97 225 103 219)(98 218 104 224)(99 223 105 217)(100 228 106 222)(101 221 107 227)(102 226 108 220)(109 240 115 234)(110 233 116 239)(111 238 117 232)(112 231 118 237)(113 236 119 230)(114 229 120 235)
G:=sub<Sym(240)| (1,56,42,30,22)(2,57,43,31,23)(3,58,44,32,24)(4,59,45,33,13)(5,60,46,34,14)(6,49,47,35,15)(7,50,48,36,16)(8,51,37,25,17)(9,52,38,26,18)(10,53,39,27,19)(11,54,40,28,20)(12,55,41,29,21)(61,111,108,90,83)(62,112,97,91,84)(63,113,98,92,73)(64,114,99,93,74)(65,115,100,94,75)(66,116,101,95,76)(67,117,102,96,77)(68,118,103,85,78)(69,119,104,86,79)(70,120,105,87,80)(71,109,106,88,81)(72,110,107,89,82)(121,175,157,154,136)(122,176,158,155,137)(123,177,159,156,138)(124,178,160,145,139)(125,179,161,146,140)(126,180,162,147,141)(127,169,163,148,142)(128,170,164,149,143)(129,171,165,150,144)(130,172,166,151,133)(131,173,167,152,134)(132,174,168,153,135)(181,235,217,205,199)(182,236,218,206,200)(183,237,219,207,201)(184,238,220,208,202)(185,239,221,209,203)(186,240,222,210,204)(187,229,223,211,193)(188,230,224,212,194)(189,231,225,213,195)(190,232,226,214,196)(191,233,227,215,197)(192,234,228,216,198), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120)(121,130,127,124)(122,131,128,125)(123,132,129,126)(133,142,139,136)(134,143,140,137)(135,144,141,138)(145,154,151,148)(146,155,152,149)(147,156,153,150)(157,166,163,160)(158,167,164,161)(159,168,165,162)(169,178,175,172)(170,179,176,173)(171,180,177,174)(181,190,187,184)(182,191,188,185)(183,192,189,186)(193,202,199,196)(194,203,200,197)(195,204,201,198)(205,214,211,208)(206,215,212,209)(207,216,213,210)(217,226,223,220)(218,227,224,221)(219,228,225,222)(229,238,235,232)(230,239,236,233)(231,240,237,234), (1,65,7,71)(2,72,8,66)(3,67,9,61)(4,62,10,68)(5,69,11,63)(6,64,12,70)(13,84,19,78)(14,79,20,73)(15,74,21,80)(16,81,22,75)(17,76,23,82)(18,83,24,77)(25,95,31,89)(26,90,32,96)(27,85,33,91)(28,92,34,86)(29,87,35,93)(30,94,36,88)(37,101,43,107)(38,108,44,102)(39,103,45,97)(40,98,46,104)(41,105,47,99)(42,100,48,106)(49,114,55,120)(50,109,56,115)(51,116,57,110)(52,111,58,117)(53,118,59,112)(54,113,60,119)(121,187,127,181)(122,182,128,188)(123,189,129,183)(124,184,130,190)(125,191,131,185)(126,186,132,192)(133,196,139,202)(134,203,140,197)(135,198,141,204)(136,193,142,199)(137,200,143,194)(138,195,144,201)(145,208,151,214)(146,215,152,209)(147,210,153,216)(148,205,154,211)(149,212,155,206)(150,207,156,213)(157,223,163,217)(158,218,164,224)(159,225,165,219)(160,220,166,226)(161,227,167,221)(162,222,168,228)(169,235,175,229)(170,230,176,236)(171,237,177,231)(172,232,178,238)(173,239,179,233)(174,234,180,240), (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), (1,123,7,129)(2,128,8,122)(3,121,9,127)(4,126,10,132)(5,131,11,125)(6,124,12,130)(13,141,19,135)(14,134,20,140)(15,139,21,133)(16,144,22,138)(17,137,23,143)(18,142,24,136)(25,155,31,149)(26,148,32,154)(27,153,33,147)(28,146,34,152)(29,151,35,145)(30,156,36,150)(37,158,43,164)(38,163,44,157)(39,168,45,162)(40,161,46,167)(41,166,47,160)(42,159,48,165)(49,178,55,172)(50,171,56,177)(51,176,57,170)(52,169,58,175)(53,174,59,180)(54,179,60,173)(61,184,67,190)(62,189,68,183)(63,182,69,188)(64,187,70,181)(65,192,71,186)(66,185,72,191)(73,200,79,194)(74,193,80,199)(75,198,81,204)(76,203,82,197)(77,196,83,202)(78,201,84,195)(85,207,91,213)(86,212,92,206)(87,205,93,211)(88,210,94,216)(89,215,95,209)(90,208,96,214)(97,225,103,219)(98,218,104,224)(99,223,105,217)(100,228,106,222)(101,221,107,227)(102,226,108,220)(109,240,115,234)(110,233,116,239)(111,238,117,232)(112,231,118,237)(113,236,119,230)(114,229,120,235)>;
G:=Group( (1,56,42,30,22)(2,57,43,31,23)(3,58,44,32,24)(4,59,45,33,13)(5,60,46,34,14)(6,49,47,35,15)(7,50,48,36,16)(8,51,37,25,17)(9,52,38,26,18)(10,53,39,27,19)(11,54,40,28,20)(12,55,41,29,21)(61,111,108,90,83)(62,112,97,91,84)(63,113,98,92,73)(64,114,99,93,74)(65,115,100,94,75)(66,116,101,95,76)(67,117,102,96,77)(68,118,103,85,78)(69,119,104,86,79)(70,120,105,87,80)(71,109,106,88,81)(72,110,107,89,82)(121,175,157,154,136)(122,176,158,155,137)(123,177,159,156,138)(124,178,160,145,139)(125,179,161,146,140)(126,180,162,147,141)(127,169,163,148,142)(128,170,164,149,143)(129,171,165,150,144)(130,172,166,151,133)(131,173,167,152,134)(132,174,168,153,135)(181,235,217,205,199)(182,236,218,206,200)(183,237,219,207,201)(184,238,220,208,202)(185,239,221,209,203)(186,240,222,210,204)(187,229,223,211,193)(188,230,224,212,194)(189,231,225,213,195)(190,232,226,214,196)(191,233,227,215,197)(192,234,228,216,198), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120)(121,130,127,124)(122,131,128,125)(123,132,129,126)(133,142,139,136)(134,143,140,137)(135,144,141,138)(145,154,151,148)(146,155,152,149)(147,156,153,150)(157,166,163,160)(158,167,164,161)(159,168,165,162)(169,178,175,172)(170,179,176,173)(171,180,177,174)(181,190,187,184)(182,191,188,185)(183,192,189,186)(193,202,199,196)(194,203,200,197)(195,204,201,198)(205,214,211,208)(206,215,212,209)(207,216,213,210)(217,226,223,220)(218,227,224,221)(219,228,225,222)(229,238,235,232)(230,239,236,233)(231,240,237,234), (1,65,7,71)(2,72,8,66)(3,67,9,61)(4,62,10,68)(5,69,11,63)(6,64,12,70)(13,84,19,78)(14,79,20,73)(15,74,21,80)(16,81,22,75)(17,76,23,82)(18,83,24,77)(25,95,31,89)(26,90,32,96)(27,85,33,91)(28,92,34,86)(29,87,35,93)(30,94,36,88)(37,101,43,107)(38,108,44,102)(39,103,45,97)(40,98,46,104)(41,105,47,99)(42,100,48,106)(49,114,55,120)(50,109,56,115)(51,116,57,110)(52,111,58,117)(53,118,59,112)(54,113,60,119)(121,187,127,181)(122,182,128,188)(123,189,129,183)(124,184,130,190)(125,191,131,185)(126,186,132,192)(133,196,139,202)(134,203,140,197)(135,198,141,204)(136,193,142,199)(137,200,143,194)(138,195,144,201)(145,208,151,214)(146,215,152,209)(147,210,153,216)(148,205,154,211)(149,212,155,206)(150,207,156,213)(157,223,163,217)(158,218,164,224)(159,225,165,219)(160,220,166,226)(161,227,167,221)(162,222,168,228)(169,235,175,229)(170,230,176,236)(171,237,177,231)(172,232,178,238)(173,239,179,233)(174,234,180,240), (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), (1,123,7,129)(2,128,8,122)(3,121,9,127)(4,126,10,132)(5,131,11,125)(6,124,12,130)(13,141,19,135)(14,134,20,140)(15,139,21,133)(16,144,22,138)(17,137,23,143)(18,142,24,136)(25,155,31,149)(26,148,32,154)(27,153,33,147)(28,146,34,152)(29,151,35,145)(30,156,36,150)(37,158,43,164)(38,163,44,157)(39,168,45,162)(40,161,46,167)(41,166,47,160)(42,159,48,165)(49,178,55,172)(50,171,56,177)(51,176,57,170)(52,169,58,175)(53,174,59,180)(54,179,60,173)(61,184,67,190)(62,189,68,183)(63,182,69,188)(64,187,70,181)(65,192,71,186)(66,185,72,191)(73,200,79,194)(74,193,80,199)(75,198,81,204)(76,203,82,197)(77,196,83,202)(78,201,84,195)(85,207,91,213)(86,212,92,206)(87,205,93,211)(88,210,94,216)(89,215,95,209)(90,208,96,214)(97,225,103,219)(98,218,104,224)(99,223,105,217)(100,228,106,222)(101,221,107,227)(102,226,108,220)(109,240,115,234)(110,233,116,239)(111,238,117,232)(112,231,118,237)(113,236,119,230)(114,229,120,235) );
G=PermutationGroup([[(1,56,42,30,22),(2,57,43,31,23),(3,58,44,32,24),(4,59,45,33,13),(5,60,46,34,14),(6,49,47,35,15),(7,50,48,36,16),(8,51,37,25,17),(9,52,38,26,18),(10,53,39,27,19),(11,54,40,28,20),(12,55,41,29,21),(61,111,108,90,83),(62,112,97,91,84),(63,113,98,92,73),(64,114,99,93,74),(65,115,100,94,75),(66,116,101,95,76),(67,117,102,96,77),(68,118,103,85,78),(69,119,104,86,79),(70,120,105,87,80),(71,109,106,88,81),(72,110,107,89,82),(121,175,157,154,136),(122,176,158,155,137),(123,177,159,156,138),(124,178,160,145,139),(125,179,161,146,140),(126,180,162,147,141),(127,169,163,148,142),(128,170,164,149,143),(129,171,165,150,144),(130,172,166,151,133),(131,173,167,152,134),(132,174,168,153,135),(181,235,217,205,199),(182,236,218,206,200),(183,237,219,207,201),(184,238,220,208,202),(185,239,221,209,203),(186,240,222,210,204),(187,229,223,211,193),(188,230,224,212,194),(189,231,225,213,195),(190,232,226,214,196),(191,233,227,215,197),(192,234,228,216,198)], [(1,4,7,10),(2,5,8,11),(3,6,9,12),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,52,55,58),(50,53,56,59),(51,54,57,60),(61,64,67,70),(62,65,68,71),(63,66,69,72),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,88,91,94),(86,89,92,95),(87,90,93,96),(97,100,103,106),(98,101,104,107),(99,102,105,108),(109,112,115,118),(110,113,116,119),(111,114,117,120),(121,130,127,124),(122,131,128,125),(123,132,129,126),(133,142,139,136),(134,143,140,137),(135,144,141,138),(145,154,151,148),(146,155,152,149),(147,156,153,150),(157,166,163,160),(158,167,164,161),(159,168,165,162),(169,178,175,172),(170,179,176,173),(171,180,177,174),(181,190,187,184),(182,191,188,185),(183,192,189,186),(193,202,199,196),(194,203,200,197),(195,204,201,198),(205,214,211,208),(206,215,212,209),(207,216,213,210),(217,226,223,220),(218,227,224,221),(219,228,225,222),(229,238,235,232),(230,239,236,233),(231,240,237,234)], [(1,65,7,71),(2,72,8,66),(3,67,9,61),(4,62,10,68),(5,69,11,63),(6,64,12,70),(13,84,19,78),(14,79,20,73),(15,74,21,80),(16,81,22,75),(17,76,23,82),(18,83,24,77),(25,95,31,89),(26,90,32,96),(27,85,33,91),(28,92,34,86),(29,87,35,93),(30,94,36,88),(37,101,43,107),(38,108,44,102),(39,103,45,97),(40,98,46,104),(41,105,47,99),(42,100,48,106),(49,114,55,120),(50,109,56,115),(51,116,57,110),(52,111,58,117),(53,118,59,112),(54,113,60,119),(121,187,127,181),(122,182,128,188),(123,189,129,183),(124,184,130,190),(125,191,131,185),(126,186,132,192),(133,196,139,202),(134,203,140,197),(135,198,141,204),(136,193,142,199),(137,200,143,194),(138,195,144,201),(145,208,151,214),(146,215,152,209),(147,210,153,216),(148,205,154,211),(149,212,155,206),(150,207,156,213),(157,223,163,217),(158,218,164,224),(159,225,165,219),(160,220,166,226),(161,227,167,221),(162,222,168,228),(169,235,175,229),(170,230,176,236),(171,237,177,231),(172,232,178,238),(173,239,179,233),(174,234,180,240)], [(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)], [(1,123,7,129),(2,128,8,122),(3,121,9,127),(4,126,10,132),(5,131,11,125),(6,124,12,130),(13,141,19,135),(14,134,20,140),(15,139,21,133),(16,144,22,138),(17,137,23,143),(18,142,24,136),(25,155,31,149),(26,148,32,154),(27,153,33,147),(28,146,34,152),(29,151,35,145),(30,156,36,150),(37,158,43,164),(38,163,44,157),(39,168,45,162),(40,161,46,167),(41,166,47,160),(42,159,48,165),(49,178,55,172),(50,171,56,177),(51,176,57,170),(52,169,58,175),(53,174,59,180),(54,179,60,173),(61,184,67,190),(62,189,68,183),(63,182,69,188),(64,187,70,181),(65,192,71,186),(66,185,72,191),(73,200,79,194),(74,193,80,199),(75,198,81,204),(76,203,82,197),(77,196,83,202),(78,201,84,195),(85,207,91,213),(86,212,92,206),(87,205,93,211),(88,210,94,216),(89,215,95,209),(90,208,96,214),(97,225,103,219),(98,218,104,224),(99,223,105,217),(100,228,106,222),(101,221,107,227),(102,226,108,220),(109,240,115,234),(110,233,116,239),(111,238,117,232),(112,231,118,237),(113,236,119,230),(114,229,120,235)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | 20R | 20S | 20T | 30A | ··· | 30L | 40A | ··· | 40H | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | ··· | 4 |
105 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×C3⋊D4 | C5×C3⋊D4 | C8.C22 | Q8.11D6 | C5×C8.C22 | C5×Q8.11D6 |
| kernel | C5×Q8.11D6 | C5×C4.Dic3 | C5×Q8⋊2S3 | C5×C3⋊Q16 | C5×C4○D12 | Q8×C30 | Q8.11D6 | C4.Dic3 | Q8⋊2S3 | C3⋊Q16 | C4○D12 | C6×Q8 | Q8×C10 | C60 | C2×C30 | C2×C20 | C5×Q8 | C20 | C2×C10 | C2×Q8 | C12 | C2×C6 | C2×C4 | Q8 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×Q8.11D6 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 0 | 0 | 0 |
| 0 | 0 | 0 | 91 | 0 | 0 |
| 0 | 0 | 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 0 | 0 | 91 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 123 | 183 | 240 | 2 |
| 0 | 0 | 153 | 30 | 240 | 1 |
| 171 | 101 | 0 | 0 | 0 | 0 |
| 140 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 222 | 19 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 |
| 0 | 0 | 73 | 138 | 38 | 203 |
| 0 | 0 | 226 | 123 | 19 | 203 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 123 | 183 | 240 | 2 |
| 0 | 0 | 211 | 153 | 240 | 1 |
| 151 | 230 | 0 | 0 | 0 | 0 |
| 79 | 90 | 0 | 0 | 0 | 0 |
| 0 | 0 | 123 | 183 | 240 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 147 | 72 | 211 | 118 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,91,0,0,0,0,0,0,91,0,0,0,0,0,0,91,0,0,0,0,0,0,91],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,123,153,0,0,1,0,183,30,0,0,0,0,240,240,0,0,0,0,2,1],[171,140,0,0,0,0,101,70,0,0,0,0,0,0,222,19,73,226,0,0,19,19,138,123,0,0,0,0,38,19,0,0,0,0,203,203],[0,240,0,0,0,0,1,240,0,0,0,0,0,0,0,1,123,211,0,0,240,0,183,153,0,0,0,0,240,240,0,0,0,0,2,1],[151,79,0,0,0,0,230,90,0,0,0,0,0,0,123,0,0,147,0,0,183,0,240,72,0,0,240,1,0,211,0,0,2,0,0,118] >;
C5×Q8.11D6 in GAP, Magma, Sage, TeX
C_5\times Q_8._{11}D_6 % in TeX
G:=Group("C5xQ8.11D6"); // GroupNames label
G:=SmallGroup(480,821);
// by ID
G=gap.SmallGroup(480,821);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,436,4204,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^5>;
// generators/relations