direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C10.D4, D6.1Dic10, Dic5⋊7(C4×S3), (S3×Dic5)⋊2C4, D6.12(C4×D5), (S3×C10).1Q8, C10.31(S3×Q8), C30.35(C2×Q8), Dic15⋊8(C2×C4), (S3×C10).29D4, C30.126(C2×D4), C10.128(S3×D4), (C2×C20).264D6, C2.4(S3×Dic10), (C2×C12).191D10, D6.17(C5⋊D4), (C2×C30).89C23, C30.51(C22×C4), C30.4Q8⋊31C2, C30.Q8⋊11C2, Dic15⋊5C4⋊16C2, C6.13(C2×Dic10), (C2×C60).408C22, (C2×Dic5).101D6, (C22×S3).85D10, (C2×Dic3).149D10, (C6×Dic5).52C22, (C2×Dic15).73C22, (C10×Dic3).177C22, C5⋊5(S3×C4⋊C4), C15⋊6(C2×C4⋊C4), C2.21(C4×S3×D5), C6.19(C2×C4×D5), C10.51(S3×C2×C4), (C5×S3)⋊2(C4⋊C4), (S3×C2×C4).11D5, C2.1(S3×C5⋊D4), (S3×C2×C20).20C2, (C2×C4).76(S3×D5), C3⋊1(C2×C10.D4), C6.29(C2×C5⋊D4), (C2×S3×Dic5).2C2, C22.44(C2×S3×D5), (C3×Dic5)⋊1(C2×C4), (S3×C10).29(C2×C4), (S3×C2×C10).82C22, (C3×C10.D4)⋊31C2, (C2×C6).101(C22×D5), (C2×C10).101(C22×S3), SmallGroup(480,475)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C10.D4
G = < a,b,c,d,e | a3=b2=c10=d4=1, e2=c5, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=d-1 >
Subgroups: 748 in 184 conjugacy classes, 74 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, Dic15, C60, S3×C10, C2×C30, C10.D4, C10.D4, C22×Dic5, C22×C20, S3×C4⋊C4, S3×Dic5, S3×Dic5, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C2×C10.D4, C30.Q8, Dic15⋊5C4, C3×C10.D4, C30.4Q8, C2×S3×Dic5, S3×C2×C20, S3×C10.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D5, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C4×S3, C22×S3, C2×C4⋊C4, Dic10, C4×D5, C5⋊D4, C22×D5, S3×C2×C4, S3×D4, S3×Q8, S3×D5, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, S3×C4⋊C4, C2×S3×D5, C2×C10.D4, S3×Dic10, C4×S3×D5, S3×C5⋊D4, S3×C10.D4
►On 240 points
Generators in S240
(1 25 11)(2 26 12)(3 27 13)(4 28 14)(5 29 15)(6 30 16)(7 21 17)(8 22 18)(9 23 19)(10 24 20)(31 237 199)(32 238 200)(33 239 191)(34 240 192)(35 231 193)(36 232 194)(37 233 195)(38 234 196)(39 235 197)(40 236 198)(41 70 52)(42 61 53)(43 62 54)(44 63 55)(45 64 56)(46 65 57)(47 66 58)(48 67 59)(49 68 60)(50 69 51)(71 107 85)(72 108 86)(73 109 87)(74 110 88)(75 101 89)(76 102 90)(77 103 81)(78 104 82)(79 105 83)(80 106 84)(91 123 116)(92 124 117)(93 125 118)(94 126 119)(95 127 120)(96 128 111)(97 129 112)(98 130 113)(99 121 114)(100 122 115)(131 186 179)(132 187 180)(133 188 171)(134 189 172)(135 190 173)(136 181 174)(137 182 175)(138 183 176)(139 184 177)(140 185 178)(141 152 164)(142 153 165)(143 154 166)(144 155 167)(145 156 168)(146 157 169)(147 158 170)(148 159 161)(149 160 162)(150 151 163)(201 212 224)(202 213 225)(203 214 226)(204 215 227)(205 216 228)(206 217 229)(207 218 230)(208 219 221)(209 220 222)(210 211 223)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 209)(32 210)(33 201)(34 202)(35 203)(36 204)(37 205)(38 206)(39 207)(40 208)(71 97)(72 98)(73 99)(74 100)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)(87 121)(88 122)(89 123)(90 124)(101 116)(102 117)(103 118)(104 119)(105 120)(106 111)(107 112)(108 113)(109 114)(110 115)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 151)(141 186)(142 187)(143 188)(144 189)(145 190)(146 181)(147 182)(148 183)(149 184)(150 185)(161 176)(162 177)(163 178)(164 179)(165 180)(166 171)(167 172)(168 173)(169 174)(170 175)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)(197 218)(198 219)(199 220)(200 211)(221 236)(222 237)(223 238)(224 239)(225 240)(226 231)(227 232)(228 233)(229 234)(230 235)
(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)
(1 173 57 163)(2 172 58 162)(3 171 59 161)(4 180 60 170)(5 179 51 169)(6 178 52 168)(7 177 53 167)(8 176 54 166)(9 175 55 165)(10 174 56 164)(11 190 65 151)(12 189 66 160)(13 188 67 159)(14 187 68 158)(15 186 69 157)(16 185 70 156)(17 184 61 155)(18 183 62 154)(19 182 63 153)(20 181 64 152)(21 139 42 144)(22 138 43 143)(23 137 44 142)(24 136 45 141)(25 135 46 150)(26 134 47 149)(27 133 48 148)(28 132 49 147)(29 131 50 146)(30 140 41 145)(31 130 215 77)(32 129 216 76)(33 128 217 75)(34 127 218 74)(35 126 219 73)(36 125 220 72)(37 124 211 71)(38 123 212 80)(39 122 213 79)(40 121 214 78)(81 199 98 204)(82 198 99 203)(83 197 100 202)(84 196 91 201)(85 195 92 210)(86 194 93 209)(87 193 94 208)(88 192 95 207)(89 191 96 206)(90 200 97 205)(101 239 111 229)(102 238 112 228)(103 237 113 227)(104 236 114 226)(105 235 115 225)(106 234 116 224)(107 233 117 223)(108 232 118 222)(109 231 119 221)(110 240 120 230)
(1 228 6 223)(2 227 7 222)(3 226 8 221)(4 225 9 230)(5 224 10 229)(11 216 16 211)(12 215 17 220)(13 214 18 219)(14 213 19 218)(15 212 20 217)(21 209 26 204)(22 208 27 203)(23 207 28 202)(24 206 29 201)(25 205 30 210)(31 61 36 66)(32 70 37 65)(33 69 38 64)(34 68 39 63)(35 67 40 62)(41 195 46 200)(42 194 47 199)(43 193 48 198)(44 192 49 197)(45 191 50 196)(51 234 56 239)(52 233 57 238)(53 232 58 237)(54 231 59 236)(55 240 60 235)(71 151 76 156)(72 160 77 155)(73 159 78 154)(74 158 79 153)(75 157 80 152)(81 144 86 149)(82 143 87 148)(83 142 88 147)(84 141 89 146)(85 150 90 145)(91 136 96 131)(92 135 97 140)(93 134 98 139)(94 133 99 138)(95 132 100 137)(101 169 106 164)(102 168 107 163)(103 167 108 162)(104 166 109 161)(105 165 110 170)(111 179 116 174)(112 178 117 173)(113 177 118 172)(114 176 119 171)(115 175 120 180)(121 183 126 188)(122 182 127 187)(123 181 128 186)(124 190 129 185)(125 189 130 184)
G:=sub<Sym(240)| (1,25,11)(2,26,12)(3,27,13)(4,28,14)(5,29,15)(6,30,16)(7,21,17)(8,22,18)(9,23,19)(10,24,20)(31,237,199)(32,238,200)(33,239,191)(34,240,192)(35,231,193)(36,232,194)(37,233,195)(38,234,196)(39,235,197)(40,236,198)(41,70,52)(42,61,53)(43,62,54)(44,63,55)(45,64,56)(46,65,57)(47,66,58)(48,67,59)(49,68,60)(50,69,51)(71,107,85)(72,108,86)(73,109,87)(74,110,88)(75,101,89)(76,102,90)(77,103,81)(78,104,82)(79,105,83)(80,106,84)(91,123,116)(92,124,117)(93,125,118)(94,126,119)(95,127,120)(96,128,111)(97,129,112)(98,130,113)(99,121,114)(100,122,115)(131,186,179)(132,187,180)(133,188,171)(134,189,172)(135,190,173)(136,181,174)(137,182,175)(138,183,176)(139,184,177)(140,185,178)(141,152,164)(142,153,165)(143,154,166)(144,155,167)(145,156,168)(146,157,169)(147,158,170)(148,159,161)(149,160,162)(150,151,163)(201,212,224)(202,213,225)(203,214,226)(204,215,227)(205,216,228)(206,217,229)(207,218,230)(208,219,221)(209,220,222)(210,211,223), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,209)(32,210)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(71,97)(72,98)(73,99)(74,100)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,121)(88,122)(89,123)(90,124)(101,116)(102,117)(103,118)(104,119)(105,120)(106,111)(107,112)(108,113)(109,114)(110,115)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,151)(141,186)(142,187)(143,188)(144,189)(145,190)(146,181)(147,182)(148,183)(149,184)(150,185)(161,176)(162,177)(163,178)(164,179)(165,180)(166,171)(167,172)(168,173)(169,174)(170,175)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(197,218)(198,219)(199,220)(200,211)(221,236)(222,237)(223,238)(224,239)(225,240)(226,231)(227,232)(228,233)(229,234)(230,235), (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), (1,173,57,163)(2,172,58,162)(3,171,59,161)(4,180,60,170)(5,179,51,169)(6,178,52,168)(7,177,53,167)(8,176,54,166)(9,175,55,165)(10,174,56,164)(11,190,65,151)(12,189,66,160)(13,188,67,159)(14,187,68,158)(15,186,69,157)(16,185,70,156)(17,184,61,155)(18,183,62,154)(19,182,63,153)(20,181,64,152)(21,139,42,144)(22,138,43,143)(23,137,44,142)(24,136,45,141)(25,135,46,150)(26,134,47,149)(27,133,48,148)(28,132,49,147)(29,131,50,146)(30,140,41,145)(31,130,215,77)(32,129,216,76)(33,128,217,75)(34,127,218,74)(35,126,219,73)(36,125,220,72)(37,124,211,71)(38,123,212,80)(39,122,213,79)(40,121,214,78)(81,199,98,204)(82,198,99,203)(83,197,100,202)(84,196,91,201)(85,195,92,210)(86,194,93,209)(87,193,94,208)(88,192,95,207)(89,191,96,206)(90,200,97,205)(101,239,111,229)(102,238,112,228)(103,237,113,227)(104,236,114,226)(105,235,115,225)(106,234,116,224)(107,233,117,223)(108,232,118,222)(109,231,119,221)(110,240,120,230), (1,228,6,223)(2,227,7,222)(3,226,8,221)(4,225,9,230)(5,224,10,229)(11,216,16,211)(12,215,17,220)(13,214,18,219)(14,213,19,218)(15,212,20,217)(21,209,26,204)(22,208,27,203)(23,207,28,202)(24,206,29,201)(25,205,30,210)(31,61,36,66)(32,70,37,65)(33,69,38,64)(34,68,39,63)(35,67,40,62)(41,195,46,200)(42,194,47,199)(43,193,48,198)(44,192,49,197)(45,191,50,196)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,151,76,156)(72,160,77,155)(73,159,78,154)(74,158,79,153)(75,157,80,152)(81,144,86,149)(82,143,87,148)(83,142,88,147)(84,141,89,146)(85,150,90,145)(91,136,96,131)(92,135,97,140)(93,134,98,139)(94,133,99,138)(95,132,100,137)(101,169,106,164)(102,168,107,163)(103,167,108,162)(104,166,109,161)(105,165,110,170)(111,179,116,174)(112,178,117,173)(113,177,118,172)(114,176,119,171)(115,175,120,180)(121,183,126,188)(122,182,127,187)(123,181,128,186)(124,190,129,185)(125,189,130,184)>;
G:=Group( (1,25,11)(2,26,12)(3,27,13)(4,28,14)(5,29,15)(6,30,16)(7,21,17)(8,22,18)(9,23,19)(10,24,20)(31,237,199)(32,238,200)(33,239,191)(34,240,192)(35,231,193)(36,232,194)(37,233,195)(38,234,196)(39,235,197)(40,236,198)(41,70,52)(42,61,53)(43,62,54)(44,63,55)(45,64,56)(46,65,57)(47,66,58)(48,67,59)(49,68,60)(50,69,51)(71,107,85)(72,108,86)(73,109,87)(74,110,88)(75,101,89)(76,102,90)(77,103,81)(78,104,82)(79,105,83)(80,106,84)(91,123,116)(92,124,117)(93,125,118)(94,126,119)(95,127,120)(96,128,111)(97,129,112)(98,130,113)(99,121,114)(100,122,115)(131,186,179)(132,187,180)(133,188,171)(134,189,172)(135,190,173)(136,181,174)(137,182,175)(138,183,176)(139,184,177)(140,185,178)(141,152,164)(142,153,165)(143,154,166)(144,155,167)(145,156,168)(146,157,169)(147,158,170)(148,159,161)(149,160,162)(150,151,163)(201,212,224)(202,213,225)(203,214,226)(204,215,227)(205,216,228)(206,217,229)(207,218,230)(208,219,221)(209,220,222)(210,211,223), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,209)(32,210)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(71,97)(72,98)(73,99)(74,100)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,121)(88,122)(89,123)(90,124)(101,116)(102,117)(103,118)(104,119)(105,120)(106,111)(107,112)(108,113)(109,114)(110,115)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,151)(141,186)(142,187)(143,188)(144,189)(145,190)(146,181)(147,182)(148,183)(149,184)(150,185)(161,176)(162,177)(163,178)(164,179)(165,180)(166,171)(167,172)(168,173)(169,174)(170,175)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(197,218)(198,219)(199,220)(200,211)(221,236)(222,237)(223,238)(224,239)(225,240)(226,231)(227,232)(228,233)(229,234)(230,235), (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), (1,173,57,163)(2,172,58,162)(3,171,59,161)(4,180,60,170)(5,179,51,169)(6,178,52,168)(7,177,53,167)(8,176,54,166)(9,175,55,165)(10,174,56,164)(11,190,65,151)(12,189,66,160)(13,188,67,159)(14,187,68,158)(15,186,69,157)(16,185,70,156)(17,184,61,155)(18,183,62,154)(19,182,63,153)(20,181,64,152)(21,139,42,144)(22,138,43,143)(23,137,44,142)(24,136,45,141)(25,135,46,150)(26,134,47,149)(27,133,48,148)(28,132,49,147)(29,131,50,146)(30,140,41,145)(31,130,215,77)(32,129,216,76)(33,128,217,75)(34,127,218,74)(35,126,219,73)(36,125,220,72)(37,124,211,71)(38,123,212,80)(39,122,213,79)(40,121,214,78)(81,199,98,204)(82,198,99,203)(83,197,100,202)(84,196,91,201)(85,195,92,210)(86,194,93,209)(87,193,94,208)(88,192,95,207)(89,191,96,206)(90,200,97,205)(101,239,111,229)(102,238,112,228)(103,237,113,227)(104,236,114,226)(105,235,115,225)(106,234,116,224)(107,233,117,223)(108,232,118,222)(109,231,119,221)(110,240,120,230), (1,228,6,223)(2,227,7,222)(3,226,8,221)(4,225,9,230)(5,224,10,229)(11,216,16,211)(12,215,17,220)(13,214,18,219)(14,213,19,218)(15,212,20,217)(21,209,26,204)(22,208,27,203)(23,207,28,202)(24,206,29,201)(25,205,30,210)(31,61,36,66)(32,70,37,65)(33,69,38,64)(34,68,39,63)(35,67,40,62)(41,195,46,200)(42,194,47,199)(43,193,48,198)(44,192,49,197)(45,191,50,196)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,151,76,156)(72,160,77,155)(73,159,78,154)(74,158,79,153)(75,157,80,152)(81,144,86,149)(82,143,87,148)(83,142,88,147)(84,141,89,146)(85,150,90,145)(91,136,96,131)(92,135,97,140)(93,134,98,139)(94,133,99,138)(95,132,100,137)(101,169,106,164)(102,168,107,163)(103,167,108,162)(104,166,109,161)(105,165,110,170)(111,179,116,174)(112,178,117,173)(113,177,118,172)(114,176,119,171)(115,175,120,180)(121,183,126,188)(122,182,127,187)(123,181,128,186)(124,190,129,185)(125,189,130,184) );
G=PermutationGroup([[(1,25,11),(2,26,12),(3,27,13),(4,28,14),(5,29,15),(6,30,16),(7,21,17),(8,22,18),(9,23,19),(10,24,20),(31,237,199),(32,238,200),(33,239,191),(34,240,192),(35,231,193),(36,232,194),(37,233,195),(38,234,196),(39,235,197),(40,236,198),(41,70,52),(42,61,53),(43,62,54),(44,63,55),(45,64,56),(46,65,57),(47,66,58),(48,67,59),(49,68,60),(50,69,51),(71,107,85),(72,108,86),(73,109,87),(74,110,88),(75,101,89),(76,102,90),(77,103,81),(78,104,82),(79,105,83),(80,106,84),(91,123,116),(92,124,117),(93,125,118),(94,126,119),(95,127,120),(96,128,111),(97,129,112),(98,130,113),(99,121,114),(100,122,115),(131,186,179),(132,187,180),(133,188,171),(134,189,172),(135,190,173),(136,181,174),(137,182,175),(138,183,176),(139,184,177),(140,185,178),(141,152,164),(142,153,165),(143,154,166),(144,155,167),(145,156,168),(146,157,169),(147,158,170),(148,159,161),(149,160,162),(150,151,163),(201,212,224),(202,213,225),(203,214,226),(204,215,227),(205,216,228),(206,217,229),(207,218,230),(208,219,221),(209,220,222),(210,211,223)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,51),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,66),(22,67),(23,68),(24,69),(25,70),(26,61),(27,62),(28,63),(29,64),(30,65),(31,209),(32,210),(33,201),(34,202),(35,203),(36,204),(37,205),(38,206),(39,207),(40,208),(71,97),(72,98),(73,99),(74,100),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96),(81,125),(82,126),(83,127),(84,128),(85,129),(86,130),(87,121),(88,122),(89,123),(90,124),(101,116),(102,117),(103,118),(104,119),(105,120),(106,111),(107,112),(108,113),(109,114),(110,115),(131,152),(132,153),(133,154),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,151),(141,186),(142,187),(143,188),(144,189),(145,190),(146,181),(147,182),(148,183),(149,184),(150,185),(161,176),(162,177),(163,178),(164,179),(165,180),(166,171),(167,172),(168,173),(169,174),(170,175),(191,212),(192,213),(193,214),(194,215),(195,216),(196,217),(197,218),(198,219),(199,220),(200,211),(221,236),(222,237),(223,238),(224,239),(225,240),(226,231),(227,232),(228,233),(229,234),(230,235)], [(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)], [(1,173,57,163),(2,172,58,162),(3,171,59,161),(4,180,60,170),(5,179,51,169),(6,178,52,168),(7,177,53,167),(8,176,54,166),(9,175,55,165),(10,174,56,164),(11,190,65,151),(12,189,66,160),(13,188,67,159),(14,187,68,158),(15,186,69,157),(16,185,70,156),(17,184,61,155),(18,183,62,154),(19,182,63,153),(20,181,64,152),(21,139,42,144),(22,138,43,143),(23,137,44,142),(24,136,45,141),(25,135,46,150),(26,134,47,149),(27,133,48,148),(28,132,49,147),(29,131,50,146),(30,140,41,145),(31,130,215,77),(32,129,216,76),(33,128,217,75),(34,127,218,74),(35,126,219,73),(36,125,220,72),(37,124,211,71),(38,123,212,80),(39,122,213,79),(40,121,214,78),(81,199,98,204),(82,198,99,203),(83,197,100,202),(84,196,91,201),(85,195,92,210),(86,194,93,209),(87,193,94,208),(88,192,95,207),(89,191,96,206),(90,200,97,205),(101,239,111,229),(102,238,112,228),(103,237,113,227),(104,236,114,226),(105,235,115,225),(106,234,116,224),(107,233,117,223),(108,232,118,222),(109,231,119,221),(110,240,120,230)], [(1,228,6,223),(2,227,7,222),(3,226,8,221),(4,225,9,230),(5,224,10,229),(11,216,16,211),(12,215,17,220),(13,214,18,219),(14,213,19,218),(15,212,20,217),(21,209,26,204),(22,208,27,203),(23,207,28,202),(24,206,29,201),(25,205,30,210),(31,61,36,66),(32,70,37,65),(33,69,38,64),(34,68,39,63),(35,67,40,62),(41,195,46,200),(42,194,47,199),(43,193,48,198),(44,192,49,197),(45,191,50,196),(51,234,56,239),(52,233,57,238),(53,232,58,237),(54,231,59,236),(55,240,60,235),(71,151,76,156),(72,160,77,155),(73,159,78,154),(74,158,79,153),(75,157,80,152),(81,144,86,149),(82,143,87,148),(83,142,88,147),(84,141,89,146),(85,150,90,145),(91,136,96,131),(92,135,97,140),(93,134,98,139),(94,133,99,138),(95,132,100,137),(101,169,106,164),(102,168,107,163),(103,167,108,162),(104,166,109,161),(105,165,110,170),(111,179,116,174),(112,178,117,173),(113,177,118,172),(114,176,119,171),(115,175,120,180),(121,183,126,188),(122,182,127,187),(123,181,128,186),(124,190,129,185),(125,189,130,184)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D5 | D6 | D6 | D10 | D10 | D10 | C4×S3 | Dic10 | C4×D5 | C5⋊D4 | S3×D4 | S3×Q8 | S3×D5 | C2×S3×D5 | S3×Dic10 | C4×S3×D5 | S3×C5⋊D4 |
| kernel | S3×C10.D4 | C30.Q8 | Dic15⋊5C4 | C3×C10.D4 | C30.4Q8 | C2×S3×Dic5 | S3×C2×C20 | S3×Dic5 | C10.D4 | S3×C10 | S3×C10 | S3×C2×C4 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | D6 | D6 | D6 | C10 | C10 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of S3×C10.D4 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 59 | 15 |
| 0 | 0 | 0 | 12 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 49 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 17 | 0 | 0 |
| 0 | 44 | 44 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 8 | 37 | 0 | 0 |
| 0 | 51 | 53 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 28 | 36 | 0 | 0 |
| 0 | 24 | 33 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,59,12,0,0,0,15,1],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,49,0,0,0,0,60],[60,0,0,0,0,0,60,44,0,0,0,17,44,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,8,51,0,0,0,37,53,0,0,0,0,0,1,0,0,0,0,0,1],[50,0,0,0,0,0,28,24,0,0,0,36,33,0,0,0,0,0,1,0,0,0,0,0,1] >;
S3×C10.D4 in GAP, Magma, Sage, TeX
S_3\times C_{10}.D_4 % in TeX
G:=Group("S3xC10.D4"); // GroupNames label
G:=SmallGroup(480,475);
// by ID
G=gap.SmallGroup(480,475);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,422,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^10=d^4=1,e^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations