direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×C4○D12, C30.87C24, C60.291C23, (C2×C20)⋊37D6, (C2×D12)⋊14C10, (C10×D12)⋊30C2, D12⋊12(C2×C10), C30⋊13(C4○D4), (C2×C60)⋊50C22, (C22×C12)⋊8C10, (C22×C20)⋊17S3, (C22×C60)⋊20C2, C6.4(C23×C10), (S3×C20)⋊25C22, (C10×Dic6)⋊31C2, Dic6⋊11(C2×C10), (C2×Dic6)⋊15C10, (C5×D12)⋊42C22, C10.72(S3×C23), C23.31(S3×C10), D6.1(C22×C10), (S3×C10).36C23, C20.238(C22×S3), C12.43(C22×C10), (C2×C30).444C23, (C5×Dic6)⋊38C22, (C22×C10).129D6, Dic3.2(C22×C10), (C5×Dic3).38C23, (C22×C30).184C22, (C10×Dic3).235C22, C6⋊1(C5×C4○D4), C3⋊1(C10×C4○D4), (S3×C2×C20)⋊31C2, (S3×C2×C4)⋊15C10, C15⋊22(C2×C4○D4), C4.43(S3×C2×C10), (C4×S3)⋊6(C2×C10), (C2×C4)⋊10(S3×C10), C3⋊D4⋊6(C2×C10), (C22×C4)⋊8(C5×S3), (C2×C12)⋊13(C2×C10), C22.6(S3×C2×C10), C2.5(S3×C22×C10), (C2×C3⋊D4)⋊12C10, (C10×C3⋊D4)⋊27C2, (C5×C3⋊D4)⋊22C22, (S3×C2×C10).120C22, (C2×C6).65(C22×C10), (C22×C6).46(C2×C10), (C22×S3).29(C2×C10), (C2×C10).257(C22×S3), (C2×Dic3).44(C2×C10), SmallGroup(480,1153)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 644 in 328 conjugacy classes, 178 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], C10, C10 [×2], C10 [×6], Dic3 [×4], C12 [×4], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×10], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C5×S3 [×4], C30, C30 [×2], C30 [×2], C2×C4○D4, C2×C20 [×2], C2×C20 [×4], C2×C20 [×10], C5×D4 [×12], C5×Q8 [×4], C22×C10, C22×C10 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×4], C60 [×4], S3×C10 [×4], S3×C10 [×4], C2×C30, C2×C30 [×2], C2×C30 [×2], C22×C20, C22×C20 [×2], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4○D12, C5×Dic6 [×4], S3×C20 [×8], C5×D12 [×4], C10×Dic3 [×2], C5×C3⋊D4 [×8], C2×C60 [×2], C2×C60 [×4], S3×C2×C10 [×2], C22×C30, C10×C4○D4, C10×Dic6, S3×C2×C20 [×2], C10×D12, C5×C4○D12 [×8], C10×C3⋊D4 [×2], C22×C60, C10×C4○D12
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C4○D4 [×2], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C2×C4○D4, C22×C10 [×15], C4○D12 [×2], S3×C23, S3×C10 [×7], C5×C4○D4 [×2], C23×C10, C2×C4○D12, S3×C2×C10 [×7], C10×C4○D4, C5×C4○D12 [×2], S3×C22×C10, C10×C4○D12
Generators and relations
G = < a,b,c,d | a10=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >
►On 240 points
Generators in S240
(1 79 46 16 71 183 33 219 205 232)(2 80 47 17 72 184 34 220 206 233)(3 81 48 18 61 185 35 221 207 234)(4 82 37 19 62 186 36 222 208 235)(5 83 38 20 63 187 25 223 209 236)(6 84 39 21 64 188 26 224 210 237)(7 73 40 22 65 189 27 225 211 238)(8 74 41 23 66 190 28 226 212 239)(9 75 42 24 67 191 29 227 213 240)(10 76 43 13 68 192 30 228 214 229)(11 77 44 14 69 181 31 217 215 230)(12 78 45 15 70 182 32 218 216 231)(49 124 88 170 139 110 152 200 105 158)(50 125 89 171 140 111 153 201 106 159)(51 126 90 172 141 112 154 202 107 160)(52 127 91 173 142 113 155 203 108 161)(53 128 92 174 143 114 156 204 97 162)(54 129 93 175 144 115 145 193 98 163)(55 130 94 176 133 116 146 194 99 164)(56 131 95 177 134 117 147 195 100 165)(57 132 96 178 135 118 148 196 101 166)(58 121 85 179 136 119 149 197 102 167)(59 122 86 180 137 120 150 198 103 168)(60 123 87 169 138 109 151 199 104 157)
(1 89 7 95)(2 90 8 96)(3 91 9 85)(4 92 10 86)(5 93 11 87)(6 94 12 88)(13 120 19 114)(14 109 20 115)(15 110 21 116)(16 111 22 117)(17 112 23 118)(18 113 24 119)(25 98 31 104)(26 99 32 105)(27 100 33 106)(28 101 34 107)(29 102 35 108)(30 103 36 97)(37 143 43 137)(38 144 44 138)(39 133 45 139)(40 134 46 140)(41 135 47 141)(42 136 48 142)(49 210 55 216)(50 211 56 205)(51 212 57 206)(52 213 58 207)(53 214 59 208)(54 215 60 209)(61 155 67 149)(62 156 68 150)(63 145 69 151)(64 146 70 152)(65 147 71 153)(66 148 72 154)(73 177 79 171)(74 178 80 172)(75 179 81 173)(76 180 82 174)(77 169 83 175)(78 170 84 176)(121 234 127 240)(122 235 128 229)(123 236 129 230)(124 237 130 231)(125 238 131 232)(126 239 132 233)(157 223 163 217)(158 224 164 218)(159 225 165 219)(160 226 166 220)(161 227 167 221)(162 228 168 222)(181 199 187 193)(182 200 188 194)(183 201 189 195)(184 202 190 196)(185 203 191 197)(186 204 192 198)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)(97 102)(98 101)(99 100)(103 108)(104 107)(105 106)(109 112)(110 111)(113 120)(114 119)(115 118)(116 117)(121 128)(122 127)(123 126)(124 125)(129 132)(130 131)(133 134)(135 144)(136 143)(137 142)(138 141)(139 140)(145 148)(146 147)(149 156)(150 155)(151 154)(152 153)(157 160)(158 159)(161 168)(162 167)(163 166)(164 165)(169 172)(170 171)(173 180)(174 179)(175 178)(176 177)(181 184)(182 183)(185 192)(186 191)(187 190)(188 189)(193 196)(194 195)(197 204)(198 203)(199 202)(200 201)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)(217 220)(218 219)(221 228)(222 227)(223 226)(224 225)(229 234)(230 233)(231 232)(235 240)(236 239)(237 238)
G:=sub<Sym(240)| (1,79,46,16,71,183,33,219,205,232)(2,80,47,17,72,184,34,220,206,233)(3,81,48,18,61,185,35,221,207,234)(4,82,37,19,62,186,36,222,208,235)(5,83,38,20,63,187,25,223,209,236)(6,84,39,21,64,188,26,224,210,237)(7,73,40,22,65,189,27,225,211,238)(8,74,41,23,66,190,28,226,212,239)(9,75,42,24,67,191,29,227,213,240)(10,76,43,13,68,192,30,228,214,229)(11,77,44,14,69,181,31,217,215,230)(12,78,45,15,70,182,32,218,216,231)(49,124,88,170,139,110,152,200,105,158)(50,125,89,171,140,111,153,201,106,159)(51,126,90,172,141,112,154,202,107,160)(52,127,91,173,142,113,155,203,108,161)(53,128,92,174,143,114,156,204,97,162)(54,129,93,175,144,115,145,193,98,163)(55,130,94,176,133,116,146,194,99,164)(56,131,95,177,134,117,147,195,100,165)(57,132,96,178,135,118,148,196,101,166)(58,121,85,179,136,119,149,197,102,167)(59,122,86,180,137,120,150,198,103,168)(60,123,87,169,138,109,151,199,104,157), (1,89,7,95)(2,90,8,96)(3,91,9,85)(4,92,10,86)(5,93,11,87)(6,94,12,88)(13,120,19,114)(14,109,20,115)(15,110,21,116)(16,111,22,117)(17,112,23,118)(18,113,24,119)(25,98,31,104)(26,99,32,105)(27,100,33,106)(28,101,34,107)(29,102,35,108)(30,103,36,97)(37,143,43,137)(38,144,44,138)(39,133,45,139)(40,134,46,140)(41,135,47,141)(42,136,48,142)(49,210,55,216)(50,211,56,205)(51,212,57,206)(52,213,58,207)(53,214,59,208)(54,215,60,209)(61,155,67,149)(62,156,68,150)(63,145,69,151)(64,146,70,152)(65,147,71,153)(66,148,72,154)(73,177,79,171)(74,178,80,172)(75,179,81,173)(76,180,82,174)(77,169,83,175)(78,170,84,176)(121,234,127,240)(122,235,128,229)(123,236,129,230)(124,237,130,231)(125,238,131,232)(126,239,132,233)(157,223,163,217)(158,224,164,218)(159,225,165,219)(160,226,166,220)(161,227,167,221)(162,228,168,222)(181,199,187,193)(182,200,188,194)(183,201,189,195)(184,202,190,196)(185,203,191,197)(186,204,192,198), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,48)(44,47)(45,46)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95)(97,102)(98,101)(99,100)(103,108)(104,107)(105,106)(109,112)(110,111)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)(129,132)(130,131)(133,134)(135,144)(136,143)(137,142)(138,141)(139,140)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(157,160)(158,159)(161,168)(162,167)(163,166)(164,165)(169,172)(170,171)(173,180)(174,179)(175,178)(176,177)(181,184)(182,183)(185,192)(186,191)(187,190)(188,189)(193,196)(194,195)(197,204)(198,203)(199,202)(200,201)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)(217,220)(218,219)(221,228)(222,227)(223,226)(224,225)(229,234)(230,233)(231,232)(235,240)(236,239)(237,238)>;
G:=Group( (1,79,46,16,71,183,33,219,205,232)(2,80,47,17,72,184,34,220,206,233)(3,81,48,18,61,185,35,221,207,234)(4,82,37,19,62,186,36,222,208,235)(5,83,38,20,63,187,25,223,209,236)(6,84,39,21,64,188,26,224,210,237)(7,73,40,22,65,189,27,225,211,238)(8,74,41,23,66,190,28,226,212,239)(9,75,42,24,67,191,29,227,213,240)(10,76,43,13,68,192,30,228,214,229)(11,77,44,14,69,181,31,217,215,230)(12,78,45,15,70,182,32,218,216,231)(49,124,88,170,139,110,152,200,105,158)(50,125,89,171,140,111,153,201,106,159)(51,126,90,172,141,112,154,202,107,160)(52,127,91,173,142,113,155,203,108,161)(53,128,92,174,143,114,156,204,97,162)(54,129,93,175,144,115,145,193,98,163)(55,130,94,176,133,116,146,194,99,164)(56,131,95,177,134,117,147,195,100,165)(57,132,96,178,135,118,148,196,101,166)(58,121,85,179,136,119,149,197,102,167)(59,122,86,180,137,120,150,198,103,168)(60,123,87,169,138,109,151,199,104,157), (1,89,7,95)(2,90,8,96)(3,91,9,85)(4,92,10,86)(5,93,11,87)(6,94,12,88)(13,120,19,114)(14,109,20,115)(15,110,21,116)(16,111,22,117)(17,112,23,118)(18,113,24,119)(25,98,31,104)(26,99,32,105)(27,100,33,106)(28,101,34,107)(29,102,35,108)(30,103,36,97)(37,143,43,137)(38,144,44,138)(39,133,45,139)(40,134,46,140)(41,135,47,141)(42,136,48,142)(49,210,55,216)(50,211,56,205)(51,212,57,206)(52,213,58,207)(53,214,59,208)(54,215,60,209)(61,155,67,149)(62,156,68,150)(63,145,69,151)(64,146,70,152)(65,147,71,153)(66,148,72,154)(73,177,79,171)(74,178,80,172)(75,179,81,173)(76,180,82,174)(77,169,83,175)(78,170,84,176)(121,234,127,240)(122,235,128,229)(123,236,129,230)(124,237,130,231)(125,238,131,232)(126,239,132,233)(157,223,163,217)(158,224,164,218)(159,225,165,219)(160,226,166,220)(161,227,167,221)(162,228,168,222)(181,199,187,193)(182,200,188,194)(183,201,189,195)(184,202,190,196)(185,203,191,197)(186,204,192,198), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,48)(44,47)(45,46)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95)(97,102)(98,101)(99,100)(103,108)(104,107)(105,106)(109,112)(110,111)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)(129,132)(130,131)(133,134)(135,144)(136,143)(137,142)(138,141)(139,140)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(157,160)(158,159)(161,168)(162,167)(163,166)(164,165)(169,172)(170,171)(173,180)(174,179)(175,178)(176,177)(181,184)(182,183)(185,192)(186,191)(187,190)(188,189)(193,196)(194,195)(197,204)(198,203)(199,202)(200,201)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)(217,220)(218,219)(221,228)(222,227)(223,226)(224,225)(229,234)(230,233)(231,232)(235,240)(236,239)(237,238) );
G=PermutationGroup([(1,79,46,16,71,183,33,219,205,232),(2,80,47,17,72,184,34,220,206,233),(3,81,48,18,61,185,35,221,207,234),(4,82,37,19,62,186,36,222,208,235),(5,83,38,20,63,187,25,223,209,236),(6,84,39,21,64,188,26,224,210,237),(7,73,40,22,65,189,27,225,211,238),(8,74,41,23,66,190,28,226,212,239),(9,75,42,24,67,191,29,227,213,240),(10,76,43,13,68,192,30,228,214,229),(11,77,44,14,69,181,31,217,215,230),(12,78,45,15,70,182,32,218,216,231),(49,124,88,170,139,110,152,200,105,158),(50,125,89,171,140,111,153,201,106,159),(51,126,90,172,141,112,154,202,107,160),(52,127,91,173,142,113,155,203,108,161),(53,128,92,174,143,114,156,204,97,162),(54,129,93,175,144,115,145,193,98,163),(55,130,94,176,133,116,146,194,99,164),(56,131,95,177,134,117,147,195,100,165),(57,132,96,178,135,118,148,196,101,166),(58,121,85,179,136,119,149,197,102,167),(59,122,86,180,137,120,150,198,103,168),(60,123,87,169,138,109,151,199,104,157)], [(1,89,7,95),(2,90,8,96),(3,91,9,85),(4,92,10,86),(5,93,11,87),(6,94,12,88),(13,120,19,114),(14,109,20,115),(15,110,21,116),(16,111,22,117),(17,112,23,118),(18,113,24,119),(25,98,31,104),(26,99,32,105),(27,100,33,106),(28,101,34,107),(29,102,35,108),(30,103,36,97),(37,143,43,137),(38,144,44,138),(39,133,45,139),(40,134,46,140),(41,135,47,141),(42,136,48,142),(49,210,55,216),(50,211,56,205),(51,212,57,206),(52,213,58,207),(53,214,59,208),(54,215,60,209),(61,155,67,149),(62,156,68,150),(63,145,69,151),(64,146,70,152),(65,147,71,153),(66,148,72,154),(73,177,79,171),(74,178,80,172),(75,179,81,173),(76,180,82,174),(77,169,83,175),(78,170,84,176),(121,234,127,240),(122,235,128,229),(123,236,129,230),(124,237,130,231),(125,238,131,232),(126,239,132,233),(157,223,163,217),(158,224,164,218),(159,225,165,219),(160,226,166,220),(161,227,167,221),(162,228,168,222),(181,199,187,193),(182,200,188,194),(183,201,189,195),(184,202,190,196),(185,203,191,197),(186,204,192,198)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,42),(38,41),(39,40),(43,48),(44,47),(45,46),(49,50),(51,60),(52,59),(53,58),(54,57),(55,56),(61,68),(62,67),(63,66),(64,65),(69,72),(70,71),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,92),(86,91),(87,90),(88,89),(93,96),(94,95),(97,102),(98,101),(99,100),(103,108),(104,107),(105,106),(109,112),(110,111),(113,120),(114,119),(115,118),(116,117),(121,128),(122,127),(123,126),(124,125),(129,132),(130,131),(133,134),(135,144),(136,143),(137,142),(138,141),(139,140),(145,148),(146,147),(149,156),(150,155),(151,154),(152,153),(157,160),(158,159),(161,168),(162,167),(163,166),(164,165),(169,172),(170,171),(173,180),(174,179),(175,178),(176,177),(181,184),(182,183),(185,192),(186,191),(187,190),(188,189),(193,196),(194,195),(197,204),(198,203),(199,202),(200,201),(205,216),(206,215),(207,214),(208,213),(209,212),(210,211),(217,220),(218,219),(221,228),(222,227),(223,226),(224,225),(229,234),(230,233),(231,232),(235,240),(236,239),(237,238)])
Matrix representation ►G ⊆ GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 60 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 46 | 23 |
| 0 | 38 | 23 |
| 60 | 0 | 0 |
| 0 | 46 | 23 |
| 0 | 38 | 15 |
G:=sub<GL(3,GF(61))| [60,0,0,0,3,0,0,0,3],[60,0,0,0,11,0,0,0,11],[1,0,0,0,46,38,0,23,23],[60,0,0,0,46,38,0,23,15] >;
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20AN | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | C4○D4 | C5×S3 | C4○D12 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C4○D12 |
| kernel | C10×C4○D12 | C10×Dic6 | S3×C2×C20 | C10×D12 | C5×C4○D12 | C10×C3⋊D4 | C22×C60 | C2×C4○D12 | C2×Dic6 | S3×C2×C4 | C2×D12 | C4○D12 | C2×C3⋊D4 | C22×C12 | C22×C20 | C2×C20 | C22×C10 | C30 | C22×C4 | C10 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 4 | 4 | 8 | 4 | 32 | 8 | 4 | 1 | 6 | 1 | 4 | 4 | 8 | 24 | 4 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times C_4\circ D_{12} % in TeX
G:=Group("C10xC4oD12"); // GroupNames label
G:=SmallGroup(480,1153);
// by ID
G=gap.SmallGroup(480,1153);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,436,2467,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀