direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8.13D6, C60.242D4, C60.234C23, D4⋊S3⋊7C10, C4○D12⋊4C10, C15⋊39(C4○D8), D4.S3⋊7C10, C3⋊Q16⋊7C10, (C5×D4).37D6, D4.8(S3×C10), C12.57(C5×D4), C6.60(D4×C10), (C2×C30).96D4, (C5×Q8).57D6, Q8⋊2S3⋊7C10, (C2×C20).369D6, C30.443(C2×D4), Q8.13(S3×C10), D12.12(C2×C10), C20.125(C3⋊D4), C12.18(C22×C10), (C2×C60).371C22, C20.207(C22×S3), Dic6.11(C2×C10), (C5×D12).51C22, (D4×C15).47C22, (Q8×C15).51C22, (C5×Dic6).53C22, C3⋊5(C5×C4○D8), (C2×C3⋊C8)⋊8C10, (C10×C3⋊C8)⋊22C2, C4○D4⋊4(C5×S3), C4.18(S3×C2×C10), (C2×C6).9(C5×D4), (C5×D4⋊S3)⋊15C2, (C5×C4○D4)⋊11S3, (C3×C4○D4)⋊2C10, C3⋊C8.10(C2×C10), C4.32(C5×C3⋊D4), (C15×C4○D4)⋊12C2, (C5×C4○D12)⋊14C2, (C2×C4).59(S3×C10), (C5×D4.S3)⋊15C2, (C5×C3⋊Q16)⋊15C2, (C3×D4).8(C2×C10), C2.24(C10×C3⋊D4), (C5×C3⋊C8).46C22, (C3×Q8).8(C2×C10), C22.1(C5×C3⋊D4), (C2×C12).45(C2×C10), (C5×Q8⋊2S3)⋊15C2, C10.145(C2×C3⋊D4), (C2×C10).22(C3⋊D4), SmallGroup(480,829)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8.13D6
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, cd=dc, ece-1=b-1c, ede-1=d5 >
Subgroups: 276 in 124 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C5×S3, C30, C30, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C5×Dic3, C60, C60, S3×C10, C2×C30, C2×C30, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C5×C4○D4, Q8.13D6, C5×C3⋊C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C4○D8, C10×C3⋊C8, C5×D4⋊S3, C5×D4.S3, C5×Q8⋊2S3, C5×C3⋊Q16, C5×C4○D12, C15×C4○D4, C5×Q8.13D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C4○D8, C5×D4, C22×C10, C2×C3⋊D4, S3×C10, D4×C10, Q8.13D6, C5×C3⋊D4, S3×C2×C10, C5×C4○D8, C10×C3⋊D4, C5×Q8.13D6
►On 240 points
Generators in S240
(1 56 47 29 18)(2 57 48 30 19)(3 58 37 31 20)(4 59 38 32 21)(5 60 39 33 22)(6 49 40 34 23)(7 50 41 35 24)(8 51 42 36 13)(9 52 43 25 14)(10 53 44 26 15)(11 54 45 27 16)(12 55 46 28 17)(61 118 108 92 73)(62 119 97 93 74)(63 120 98 94 75)(64 109 99 95 76)(65 110 100 96 77)(66 111 101 85 78)(67 112 102 86 79)(68 113 103 87 80)(69 114 104 88 81)(70 115 105 89 82)(71 116 106 90 83)(72 117 107 91 84)(121 178 160 148 133)(122 179 161 149 134)(123 180 162 150 135)(124 169 163 151 136)(125 170 164 152 137)(126 171 165 153 138)(127 172 166 154 139)(128 173 167 155 140)(129 174 168 156 141)(130 175 157 145 142)(131 176 158 146 143)(132 177 159 147 144)(181 235 223 208 199)(182 236 224 209 200)(183 237 225 210 201)(184 238 226 211 202)(185 239 227 212 203)(186 240 228 213 204)(187 229 217 214 193)(188 230 218 215 194)(189 231 219 216 195)(190 232 220 205 196)(191 233 221 206 197)(192 234 222 207 198)
(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 70 67 64)(62 71 68 65)(63 72 69 66)(73 82 79 76)(74 83 80 77)(75 84 81 78)(85 94 91 88)(86 95 92 89)(87 96 93 90)(97 106 103 100)(98 107 104 101)(99 108 105 102)(109 118 115 112)(110 119 116 113)(111 120 117 114)(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)(181 184 187 190)(182 185 188 191)(183 186 189 192)(193 196 199 202)(194 197 200 203)(195 198 201 204)(205 208 211 214)(206 209 212 215)(207 210 213 216)(217 220 223 226)(218 221 224 227)(219 222 225 228)(229 232 235 238)(230 233 236 239)(231 234 237 240)
(1 68 7 62)(2 69 8 63)(3 70 9 64)(4 71 10 65)(5 72 11 66)(6 61 12 67)(13 75 19 81)(14 76 20 82)(15 77 21 83)(16 78 22 84)(17 79 23 73)(18 80 24 74)(25 95 31 89)(26 96 32 90)(27 85 33 91)(28 86 34 92)(29 87 35 93)(30 88 36 94)(37 105 43 99)(38 106 44 100)(39 107 45 101)(40 108 46 102)(41 97 47 103)(42 98 48 104)(49 118 55 112)(50 119 56 113)(51 120 57 114)(52 109 58 115)(53 110 59 116)(54 111 60 117)(121 190 127 184)(122 191 128 185)(123 192 129 186)(124 181 130 187)(125 182 131 188)(126 183 132 189)(133 196 139 202)(134 197 140 203)(135 198 141 204)(136 199 142 193)(137 200 143 194)(138 201 144 195)(145 214 151 208)(146 215 152 209)(147 216 153 210)(148 205 154 211)(149 206 155 212)(150 207 156 213)(157 217 163 223)(158 218 164 224)(159 219 165 225)(160 220 166 226)(161 221 167 227)(162 222 168 228)(169 235 175 229)(170 236 176 230)(171 237 177 231)(172 238 178 232)(173 239 179 233)(174 240 180 234)
(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)
(1 132 7 126)(2 125 8 131)(3 130 9 124)(4 123 10 129)(5 128 11 122)(6 121 12 127)(13 143 19 137)(14 136 20 142)(15 141 21 135)(16 134 22 140)(17 139 23 133)(18 144 24 138)(25 151 31 145)(26 156 32 150)(27 149 33 155)(28 154 34 148)(29 147 35 153)(30 152 36 146)(37 157 43 163)(38 162 44 168)(39 167 45 161)(40 160 46 166)(41 165 47 159)(42 158 48 164)(49 178 55 172)(50 171 56 177)(51 176 57 170)(52 169 58 175)(53 174 59 180)(54 179 60 173)(61 181 67 187)(62 186 68 192)(63 191 69 185)(64 184 70 190)(65 189 71 183)(66 182 72 188)(73 199 79 193)(74 204 80 198)(75 197 81 203)(76 202 82 196)(77 195 83 201)(78 200 84 194)(85 209 91 215)(86 214 92 208)(87 207 93 213)(88 212 94 206)(89 205 95 211)(90 210 96 216)(97 228 103 222)(98 221 104 227)(99 226 105 220)(100 219 106 225)(101 224 107 218)(102 217 108 223)(109 238 115 232)(110 231 116 237)(111 236 117 230)(112 229 118 235)(113 234 119 240)(114 239 120 233)
G:=sub<Sym(240)| (1,56,47,29,18)(2,57,48,30,19)(3,58,37,31,20)(4,59,38,32,21)(5,60,39,33,22)(6,49,40,34,23)(7,50,41,35,24)(8,51,42,36,13)(9,52,43,25,14)(10,53,44,26,15)(11,54,45,27,16)(12,55,46,28,17)(61,118,108,92,73)(62,119,97,93,74)(63,120,98,94,75)(64,109,99,95,76)(65,110,100,96,77)(66,111,101,85,78)(67,112,102,86,79)(68,113,103,87,80)(69,114,104,88,81)(70,115,105,89,82)(71,116,106,90,83)(72,117,107,91,84)(121,178,160,148,133)(122,179,161,149,134)(123,180,162,150,135)(124,169,163,151,136)(125,170,164,152,137)(126,171,165,153,138)(127,172,166,154,139)(128,173,167,155,140)(129,174,168,156,141)(130,175,157,145,142)(131,176,158,146,143)(132,177,159,147,144)(181,235,223,208,199)(182,236,224,209,200)(183,237,225,210,201)(184,238,226,211,202)(185,239,227,212,203)(186,240,228,213,204)(187,229,217,214,193)(188,230,218,215,194)(189,231,219,216,195)(190,232,220,205,196)(191,233,221,206,197)(192,234,222,207,198), (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)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,82,79,76)(74,83,80,77)(75,84,81,78)(85,94,91,88)(86,95,92,89)(87,96,93,90)(97,106,103,100)(98,107,104,101)(99,108,105,102)(109,118,115,112)(110,119,116,113)(111,120,117,114)(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)(181,184,187,190)(182,185,188,191)(183,186,189,192)(193,196,199,202)(194,197,200,203)(195,198,201,204)(205,208,211,214)(206,209,212,215)(207,210,213,216)(217,220,223,226)(218,221,224,227)(219,222,225,228)(229,232,235,238)(230,233,236,239)(231,234,237,240), (1,68,7,62)(2,69,8,63)(3,70,9,64)(4,71,10,65)(5,72,11,66)(6,61,12,67)(13,75,19,81)(14,76,20,82)(15,77,21,83)(16,78,22,84)(17,79,23,73)(18,80,24,74)(25,95,31,89)(26,96,32,90)(27,85,33,91)(28,86,34,92)(29,87,35,93)(30,88,36,94)(37,105,43,99)(38,106,44,100)(39,107,45,101)(40,108,46,102)(41,97,47,103)(42,98,48,104)(49,118,55,112)(50,119,56,113)(51,120,57,114)(52,109,58,115)(53,110,59,116)(54,111,60,117)(121,190,127,184)(122,191,128,185)(123,192,129,186)(124,181,130,187)(125,182,131,188)(126,183,132,189)(133,196,139,202)(134,197,140,203)(135,198,141,204)(136,199,142,193)(137,200,143,194)(138,201,144,195)(145,214,151,208)(146,215,152,209)(147,216,153,210)(148,205,154,211)(149,206,155,212)(150,207,156,213)(157,217,163,223)(158,218,164,224)(159,219,165,225)(160,220,166,226)(161,221,167,227)(162,222,168,228)(169,235,175,229)(170,236,176,230)(171,237,177,231)(172,238,178,232)(173,239,179,233)(174,240,180,234), (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), (1,132,7,126)(2,125,8,131)(3,130,9,124)(4,123,10,129)(5,128,11,122)(6,121,12,127)(13,143,19,137)(14,136,20,142)(15,141,21,135)(16,134,22,140)(17,139,23,133)(18,144,24,138)(25,151,31,145)(26,156,32,150)(27,149,33,155)(28,154,34,148)(29,147,35,153)(30,152,36,146)(37,157,43,163)(38,162,44,168)(39,167,45,161)(40,160,46,166)(41,165,47,159)(42,158,48,164)(49,178,55,172)(50,171,56,177)(51,176,57,170)(52,169,58,175)(53,174,59,180)(54,179,60,173)(61,181,67,187)(62,186,68,192)(63,191,69,185)(64,184,70,190)(65,189,71,183)(66,182,72,188)(73,199,79,193)(74,204,80,198)(75,197,81,203)(76,202,82,196)(77,195,83,201)(78,200,84,194)(85,209,91,215)(86,214,92,208)(87,207,93,213)(88,212,94,206)(89,205,95,211)(90,210,96,216)(97,228,103,222)(98,221,104,227)(99,226,105,220)(100,219,106,225)(101,224,107,218)(102,217,108,223)(109,238,115,232)(110,231,116,237)(111,236,117,230)(112,229,118,235)(113,234,119,240)(114,239,120,233)>;
G:=Group( (1,56,47,29,18)(2,57,48,30,19)(3,58,37,31,20)(4,59,38,32,21)(5,60,39,33,22)(6,49,40,34,23)(7,50,41,35,24)(8,51,42,36,13)(9,52,43,25,14)(10,53,44,26,15)(11,54,45,27,16)(12,55,46,28,17)(61,118,108,92,73)(62,119,97,93,74)(63,120,98,94,75)(64,109,99,95,76)(65,110,100,96,77)(66,111,101,85,78)(67,112,102,86,79)(68,113,103,87,80)(69,114,104,88,81)(70,115,105,89,82)(71,116,106,90,83)(72,117,107,91,84)(121,178,160,148,133)(122,179,161,149,134)(123,180,162,150,135)(124,169,163,151,136)(125,170,164,152,137)(126,171,165,153,138)(127,172,166,154,139)(128,173,167,155,140)(129,174,168,156,141)(130,175,157,145,142)(131,176,158,146,143)(132,177,159,147,144)(181,235,223,208,199)(182,236,224,209,200)(183,237,225,210,201)(184,238,226,211,202)(185,239,227,212,203)(186,240,228,213,204)(187,229,217,214,193)(188,230,218,215,194)(189,231,219,216,195)(190,232,220,205,196)(191,233,221,206,197)(192,234,222,207,198), (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)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,82,79,76)(74,83,80,77)(75,84,81,78)(85,94,91,88)(86,95,92,89)(87,96,93,90)(97,106,103,100)(98,107,104,101)(99,108,105,102)(109,118,115,112)(110,119,116,113)(111,120,117,114)(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)(181,184,187,190)(182,185,188,191)(183,186,189,192)(193,196,199,202)(194,197,200,203)(195,198,201,204)(205,208,211,214)(206,209,212,215)(207,210,213,216)(217,220,223,226)(218,221,224,227)(219,222,225,228)(229,232,235,238)(230,233,236,239)(231,234,237,240), (1,68,7,62)(2,69,8,63)(3,70,9,64)(4,71,10,65)(5,72,11,66)(6,61,12,67)(13,75,19,81)(14,76,20,82)(15,77,21,83)(16,78,22,84)(17,79,23,73)(18,80,24,74)(25,95,31,89)(26,96,32,90)(27,85,33,91)(28,86,34,92)(29,87,35,93)(30,88,36,94)(37,105,43,99)(38,106,44,100)(39,107,45,101)(40,108,46,102)(41,97,47,103)(42,98,48,104)(49,118,55,112)(50,119,56,113)(51,120,57,114)(52,109,58,115)(53,110,59,116)(54,111,60,117)(121,190,127,184)(122,191,128,185)(123,192,129,186)(124,181,130,187)(125,182,131,188)(126,183,132,189)(133,196,139,202)(134,197,140,203)(135,198,141,204)(136,199,142,193)(137,200,143,194)(138,201,144,195)(145,214,151,208)(146,215,152,209)(147,216,153,210)(148,205,154,211)(149,206,155,212)(150,207,156,213)(157,217,163,223)(158,218,164,224)(159,219,165,225)(160,220,166,226)(161,221,167,227)(162,222,168,228)(169,235,175,229)(170,236,176,230)(171,237,177,231)(172,238,178,232)(173,239,179,233)(174,240,180,234), (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), (1,132,7,126)(2,125,8,131)(3,130,9,124)(4,123,10,129)(5,128,11,122)(6,121,12,127)(13,143,19,137)(14,136,20,142)(15,141,21,135)(16,134,22,140)(17,139,23,133)(18,144,24,138)(25,151,31,145)(26,156,32,150)(27,149,33,155)(28,154,34,148)(29,147,35,153)(30,152,36,146)(37,157,43,163)(38,162,44,168)(39,167,45,161)(40,160,46,166)(41,165,47,159)(42,158,48,164)(49,178,55,172)(50,171,56,177)(51,176,57,170)(52,169,58,175)(53,174,59,180)(54,179,60,173)(61,181,67,187)(62,186,68,192)(63,191,69,185)(64,184,70,190)(65,189,71,183)(66,182,72,188)(73,199,79,193)(74,204,80,198)(75,197,81,203)(76,202,82,196)(77,195,83,201)(78,200,84,194)(85,209,91,215)(86,214,92,208)(87,207,93,213)(88,212,94,206)(89,205,95,211)(90,210,96,216)(97,228,103,222)(98,221,104,227)(99,226,105,220)(100,219,106,225)(101,224,107,218)(102,217,108,223)(109,238,115,232)(110,231,116,237)(111,236,117,230)(112,229,118,235)(113,234,119,240)(114,239,120,233) );
G=PermutationGroup([[(1,56,47,29,18),(2,57,48,30,19),(3,58,37,31,20),(4,59,38,32,21),(5,60,39,33,22),(6,49,40,34,23),(7,50,41,35,24),(8,51,42,36,13),(9,52,43,25,14),(10,53,44,26,15),(11,54,45,27,16),(12,55,46,28,17),(61,118,108,92,73),(62,119,97,93,74),(63,120,98,94,75),(64,109,99,95,76),(65,110,100,96,77),(66,111,101,85,78),(67,112,102,86,79),(68,113,103,87,80),(69,114,104,88,81),(70,115,105,89,82),(71,116,106,90,83),(72,117,107,91,84),(121,178,160,148,133),(122,179,161,149,134),(123,180,162,150,135),(124,169,163,151,136),(125,170,164,152,137),(126,171,165,153,138),(127,172,166,154,139),(128,173,167,155,140),(129,174,168,156,141),(130,175,157,145,142),(131,176,158,146,143),(132,177,159,147,144),(181,235,223,208,199),(182,236,224,209,200),(183,237,225,210,201),(184,238,226,211,202),(185,239,227,212,203),(186,240,228,213,204),(187,229,217,214,193),(188,230,218,215,194),(189,231,219,216,195),(190,232,220,205,196),(191,233,221,206,197),(192,234,222,207,198)], [(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),(61,70,67,64),(62,71,68,65),(63,72,69,66),(73,82,79,76),(74,83,80,77),(75,84,81,78),(85,94,91,88),(86,95,92,89),(87,96,93,90),(97,106,103,100),(98,107,104,101),(99,108,105,102),(109,118,115,112),(110,119,116,113),(111,120,117,114),(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),(181,184,187,190),(182,185,188,191),(183,186,189,192),(193,196,199,202),(194,197,200,203),(195,198,201,204),(205,208,211,214),(206,209,212,215),(207,210,213,216),(217,220,223,226),(218,221,224,227),(219,222,225,228),(229,232,235,238),(230,233,236,239),(231,234,237,240)], [(1,68,7,62),(2,69,8,63),(3,70,9,64),(4,71,10,65),(5,72,11,66),(6,61,12,67),(13,75,19,81),(14,76,20,82),(15,77,21,83),(16,78,22,84),(17,79,23,73),(18,80,24,74),(25,95,31,89),(26,96,32,90),(27,85,33,91),(28,86,34,92),(29,87,35,93),(30,88,36,94),(37,105,43,99),(38,106,44,100),(39,107,45,101),(40,108,46,102),(41,97,47,103),(42,98,48,104),(49,118,55,112),(50,119,56,113),(51,120,57,114),(52,109,58,115),(53,110,59,116),(54,111,60,117),(121,190,127,184),(122,191,128,185),(123,192,129,186),(124,181,130,187),(125,182,131,188),(126,183,132,189),(133,196,139,202),(134,197,140,203),(135,198,141,204),(136,199,142,193),(137,200,143,194),(138,201,144,195),(145,214,151,208),(146,215,152,209),(147,216,153,210),(148,205,154,211),(149,206,155,212),(150,207,156,213),(157,217,163,223),(158,218,164,224),(159,219,165,225),(160,220,166,226),(161,221,167,227),(162,222,168,228),(169,235,175,229),(170,236,176,230),(171,237,177,231),(172,238,178,232),(173,239,179,233),(174,240,180,234)], [(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)], [(1,132,7,126),(2,125,8,131),(3,130,9,124),(4,123,10,129),(5,128,11,122),(6,121,12,127),(13,143,19,137),(14,136,20,142),(15,141,21,135),(16,134,22,140),(17,139,23,133),(18,144,24,138),(25,151,31,145),(26,156,32,150),(27,149,33,155),(28,154,34,148),(29,147,35,153),(30,152,36,146),(37,157,43,163),(38,162,44,168),(39,167,45,161),(40,160,46,166),(41,165,47,159),(42,158,48,164),(49,178,55,172),(50,171,56,177),(51,176,57,170),(52,169,58,175),(53,174,59,180),(54,179,60,173),(61,181,67,187),(62,186,68,192),(63,191,69,185),(64,184,70,190),(65,189,71,183),(66,182,72,188),(73,199,79,193),(74,204,80,198),(75,197,81,203),(76,202,82,196),(77,195,83,201),(78,200,84,194),(85,209,91,215),(86,214,92,208),(87,207,93,213),(88,212,94,206),(89,205,95,211),(90,210,96,216),(97,228,103,222),(98,221,104,227),(99,226,105,220),(100,219,106,225),(101,224,107,218),(102,217,108,223),(109,238,115,232),(110,231,116,237),(111,236,117,230),(112,229,118,235),(113,234,119,240),(114,239,120,233)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 10O | 10P | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | 20N | 20O | 20P | 20Q | 20R | 20S | 20T | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 40A | ··· | 40P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 12 | 2 | 1 | 1 | 2 | 4 | 12 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C5×S3 | C4○D8 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | S3×C10 | C5×C3⋊D4 | C5×C3⋊D4 | C5×C4○D8 | Q8.13D6 | C5×Q8.13D6 |
| kernel | C5×Q8.13D6 | C10×C3⋊C8 | C5×D4⋊S3 | C5×D4.S3 | C5×Q8⋊2S3 | C5×C3⋊Q16 | C5×C4○D12 | C15×C4○D4 | Q8.13D6 | C2×C3⋊C8 | D4⋊S3 | D4.S3 | Q8⋊2S3 | C3⋊Q16 | C4○D12 | C3×C4○D4 | C5×C4○D4 | C60 | C2×C30 | C2×C20 | C5×D4 | C5×Q8 | C20 | C2×C10 | C4○D4 | C15 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C3 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 2 | 8 |
Matrix representation of C5×Q8.13D6 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 64 | 177 |
| 171 | 101 | 0 | 0 |
| 140 | 70 | 0 | 0 |
| 0 | 0 | 177 | 128 |
| 0 | 0 | 0 | 64 |
| 0 | 240 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 239 | 202 | 0 | 0 |
| 204 | 2 | 0 | 0 |
| 0 | 0 | 211 | 60 |
| 0 | 0 | 230 | 30 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,98,0,0,0,0,98],[240,0,0,0,0,240,0,0,0,0,64,64,0,0,0,177],[171,140,0,0,101,70,0,0,0,0,177,0,0,0,128,64],[0,1,0,0,240,1,0,0,0,0,177,0,0,0,0,177],[239,204,0,0,202,2,0,0,0,0,211,230,0,0,60,30] >;
C5×Q8.13D6 in GAP, Magma, Sage, TeX
C_5\times Q_8._{13}D_6 % in TeX
G:=Group("C5xQ8.13D6"); // GroupNames label
G:=SmallGroup(480,829);
// by ID
G=gap.SmallGroup(480,829);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,4204,1068,102,15686]);
// 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,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations