direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×Q8×C10, C30.91C24, C60.238C23, C6⋊2(Q8×C10), (C6×Q8)⋊5C10, C30⋊10(C2×Q8), (Q8×C30)⋊19C2, C15⋊11(C22×Q8), Dic6⋊9(C2×C10), (C2×C20).371D6, C6.8(C23×C10), (C10×Dic6)⋊29C2, (C2×Dic6)⋊13C10, C10.76(S3×C23), (Q8×C15)⋊32C22, D6.9(C22×C10), (S3×C10).45C23, (S3×C20).62C22, (C2×C30).446C23, C12.22(C22×C10), (C2×C60).374C22, C20.211(C22×S3), (C5×Dic6)⋊36C22, (C5×Dic3).41C23, Dic3.5(C22×C10), (C10×Dic3).236C22, C3⋊2(Q8×C2×C10), (S3×C2×C4).6C10, C4.22(S3×C2×C10), (S3×C2×C20).17C2, (C3×Q8)⋊5(C2×C10), C2.9(S3×C22×C10), (C2×C4).61(S3×C10), C22.31(S3×C2×C10), (C4×S3).13(C2×C10), (C2×C12).48(C2×C10), (S3×C2×C10).128C22, (C2×C6).66(C22×C10), (C22×S3).36(C2×C10), (C2×C10).378(C22×S3), (C2×Dic3).45(C2×C10), SmallGroup(480,1157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Q8×C10
G = < a,b,c,d,e | a10=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 548 in 312 conjugacy classes, 194 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, Q8, Q8, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C30, C22×Q8, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×C10, C2×Dic6, S3×C2×C4, S3×Q8, C6×Q8, C5×Dic3, C60, S3×C10, C2×C30, C22×C20, Q8×C10, Q8×C10, C2×S3×Q8, C5×Dic6, S3×C20, C10×Dic3, C2×C60, Q8×C15, S3×C2×C10, Q8×C2×C10, C10×Dic6, S3×C2×C20, C5×S3×Q8, Q8×C30, S3×Q8×C10
Quotients: C1, C2, C22, C5, S3, Q8, C23, C10, D6, C2×Q8, C24, C2×C10, C22×S3, C5×S3, C22×Q8, C5×Q8, C22×C10, S3×Q8, S3×C23, S3×C10, Q8×C10, C23×C10, C2×S3×Q8, S3×C2×C10, Q8×C2×C10, C5×S3×Q8, S3×C22×C10, S3×Q8×C10
►On 240 points
Generators in S240
(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 97 191)(2 98 192)(3 99 193)(4 100 194)(5 91 195)(6 92 196)(7 93 197)(8 94 198)(9 95 199)(10 96 200)(11 179 161)(12 180 162)(13 171 163)(14 172 164)(15 173 165)(16 174 166)(17 175 167)(18 176 168)(19 177 169)(20 178 170)(21 80 135)(22 71 136)(23 72 137)(24 73 138)(25 74 139)(26 75 140)(27 76 131)(28 77 132)(29 78 133)(30 79 134)(31 67 211)(32 68 212)(33 69 213)(34 70 214)(35 61 215)(36 62 216)(37 63 217)(38 64 218)(39 65 219)(40 66 220)(41 105 225)(42 106 226)(43 107 227)(44 108 228)(45 109 229)(46 110 230)(47 101 221)(48 102 222)(49 103 223)(50 104 224)(51 115 209)(52 116 210)(53 117 201)(54 118 202)(55 119 203)(56 120 204)(57 111 205)(58 112 206)(59 113 207)(60 114 208)(81 126 239)(82 127 240)(83 128 231)(84 129 232)(85 130 233)(86 121 234)(87 122 235)(88 123 236)(89 124 237)(90 125 238)(141 160 188)(142 151 189)(143 152 190)(144 153 181)(145 154 182)(146 155 183)(147 156 184)(148 157 185)(149 158 186)(150 159 187)
(1 202)(2 203)(3 204)(4 205)(5 206)(6 207)(7 208)(8 209)(9 210)(10 201)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 131)(21 164)(22 165)(23 166)(24 167)(25 168)(26 169)(27 170)(28 161)(29 162)(30 163)(31 50)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(51 198)(52 199)(53 200)(54 191)(55 192)(56 193)(57 194)(58 195)(59 196)(60 197)(61 228)(62 229)(63 230)(64 221)(65 222)(66 223)(67 224)(68 225)(69 226)(70 227)(71 173)(72 174)(73 175)(74 176)(75 177)(76 178)(77 179)(78 180)(79 171)(80 172)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 151)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 111)(101 218)(102 219)(103 220)(104 211)(105 212)(106 213)(107 214)(108 215)(109 216)(110 217)(121 148)(122 149)(123 150)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(181 240)(182 231)(183 232)(184 233)(185 234)(186 235)(187 236)(188 237)(189 238)(190 239)
(1 237 207 183)(2 238 208 184)(3 239 209 185)(4 240 210 186)(5 231 201 187)(6 232 202 188)(7 233 203 189)(8 234 204 190)(9 235 205 181)(10 236 206 182)(11 218 72 42)(12 219 73 43)(13 220 74 44)(14 211 75 45)(15 212 76 46)(16 213 77 47)(17 214 78 48)(18 215 79 49)(19 216 80 50)(20 217 71 41)(21 224 169 62)(22 225 170 63)(23 226 161 64)(24 227 162 65)(25 228 163 66)(26 229 164 67)(27 230 165 68)(28 221 166 69)(29 222 167 70)(30 223 168 61)(31 140 109 172)(32 131 110 173)(33 132 101 174)(34 133 102 175)(35 134 103 176)(36 135 104 177)(37 136 105 178)(38 137 106 179)(39 138 107 180)(40 139 108 171)(51 148 99 81)(52 149 100 82)(53 150 91 83)(54 141 92 84)(55 142 93 85)(56 143 94 86)(57 144 95 87)(58 145 96 88)(59 146 97 89)(60 147 98 90)(111 153 199 122)(112 154 200 123)(113 155 191 124)(114 156 192 125)(115 157 193 126)(116 158 194 127)(117 159 195 128)(118 160 196 129)(119 151 197 130)(120 152 198 121)
(1 230 207 68)(2 221 208 69)(3 222 209 70)(4 223 210 61)(5 224 201 62)(6 225 202 63)(7 226 203 64)(8 227 204 65)(9 228 205 66)(10 229 206 67)(11 85 72 142)(12 86 73 143)(13 87 74 144)(14 88 75 145)(15 89 76 146)(16 90 77 147)(17 81 78 148)(18 82 79 149)(19 83 80 150)(20 84 71 141)(21 187 169 231)(22 188 170 232)(23 189 161 233)(24 190 162 234)(25 181 163 235)(26 182 164 236)(27 183 165 237)(28 184 166 238)(29 185 167 239)(30 186 168 240)(31 200 109 112)(32 191 110 113)(33 192 101 114)(34 193 102 115)(35 194 103 116)(36 195 104 117)(37 196 105 118)(38 197 106 119)(39 198 107 120)(40 199 108 111)(41 54 217 92)(42 55 218 93)(43 56 219 94)(44 57 220 95)(45 58 211 96)(46 59 212 97)(47 60 213 98)(48 51 214 99)(49 52 215 100)(50 53 216 91)(121 138 152 180)(122 139 153 171)(123 140 154 172)(124 131 155 173)(125 132 156 174)(126 133 157 175)(127 134 158 176)(128 135 159 177)(129 136 160 178)(130 137 151 179)
G:=sub<Sym(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), (1,97,191)(2,98,192)(3,99,193)(4,100,194)(5,91,195)(6,92,196)(7,93,197)(8,94,198)(9,95,199)(10,96,200)(11,179,161)(12,180,162)(13,171,163)(14,172,164)(15,173,165)(16,174,166)(17,175,167)(18,176,168)(19,177,169)(20,178,170)(21,80,135)(22,71,136)(23,72,137)(24,73,138)(25,74,139)(26,75,140)(27,76,131)(28,77,132)(29,78,133)(30,79,134)(31,67,211)(32,68,212)(33,69,213)(34,70,214)(35,61,215)(36,62,216)(37,63,217)(38,64,218)(39,65,219)(40,66,220)(41,105,225)(42,106,226)(43,107,227)(44,108,228)(45,109,229)(46,110,230)(47,101,221)(48,102,222)(49,103,223)(50,104,224)(51,115,209)(52,116,210)(53,117,201)(54,118,202)(55,119,203)(56,120,204)(57,111,205)(58,112,206)(59,113,207)(60,114,208)(81,126,239)(82,127,240)(83,128,231)(84,129,232)(85,130,233)(86,121,234)(87,122,235)(88,123,236)(89,124,237)(90,125,238)(141,160,188)(142,151,189)(143,152,190)(144,153,181)(145,154,182)(146,155,183)(147,156,184)(148,157,185)(149,158,186)(150,159,187), (1,202)(2,203)(3,204)(4,205)(5,206)(6,207)(7,208)(8,209)(9,210)(10,201)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,131)(21,164)(22,165)(23,166)(24,167)(25,168)(26,169)(27,170)(28,161)(29,162)(30,163)(31,50)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(51,198)(52,199)(53,200)(54,191)(55,192)(56,193)(57,194)(58,195)(59,196)(60,197)(61,228)(62,229)(63,230)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,173)(72,174)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,171)(80,172)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,151)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,111)(101,218)(102,219)(103,220)(104,211)(105,212)(106,213)(107,214)(108,215)(109,216)(110,217)(121,148)(122,149)(123,150)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(181,240)(182,231)(183,232)(184,233)(185,234)(186,235)(187,236)(188,237)(189,238)(190,239), (1,237,207,183)(2,238,208,184)(3,239,209,185)(4,240,210,186)(5,231,201,187)(6,232,202,188)(7,233,203,189)(8,234,204,190)(9,235,205,181)(10,236,206,182)(11,218,72,42)(12,219,73,43)(13,220,74,44)(14,211,75,45)(15,212,76,46)(16,213,77,47)(17,214,78,48)(18,215,79,49)(19,216,80,50)(20,217,71,41)(21,224,169,62)(22,225,170,63)(23,226,161,64)(24,227,162,65)(25,228,163,66)(26,229,164,67)(27,230,165,68)(28,221,166,69)(29,222,167,70)(30,223,168,61)(31,140,109,172)(32,131,110,173)(33,132,101,174)(34,133,102,175)(35,134,103,176)(36,135,104,177)(37,136,105,178)(38,137,106,179)(39,138,107,180)(40,139,108,171)(51,148,99,81)(52,149,100,82)(53,150,91,83)(54,141,92,84)(55,142,93,85)(56,143,94,86)(57,144,95,87)(58,145,96,88)(59,146,97,89)(60,147,98,90)(111,153,199,122)(112,154,200,123)(113,155,191,124)(114,156,192,125)(115,157,193,126)(116,158,194,127)(117,159,195,128)(118,160,196,129)(119,151,197,130)(120,152,198,121), (1,230,207,68)(2,221,208,69)(3,222,209,70)(4,223,210,61)(5,224,201,62)(6,225,202,63)(7,226,203,64)(8,227,204,65)(9,228,205,66)(10,229,206,67)(11,85,72,142)(12,86,73,143)(13,87,74,144)(14,88,75,145)(15,89,76,146)(16,90,77,147)(17,81,78,148)(18,82,79,149)(19,83,80,150)(20,84,71,141)(21,187,169,231)(22,188,170,232)(23,189,161,233)(24,190,162,234)(25,181,163,235)(26,182,164,236)(27,183,165,237)(28,184,166,238)(29,185,167,239)(30,186,168,240)(31,200,109,112)(32,191,110,113)(33,192,101,114)(34,193,102,115)(35,194,103,116)(36,195,104,117)(37,196,105,118)(38,197,106,119)(39,198,107,120)(40,199,108,111)(41,54,217,92)(42,55,218,93)(43,56,219,94)(44,57,220,95)(45,58,211,96)(46,59,212,97)(47,60,213,98)(48,51,214,99)(49,52,215,100)(50,53,216,91)(121,138,152,180)(122,139,153,171)(123,140,154,172)(124,131,155,173)(125,132,156,174)(126,133,157,175)(127,134,158,176)(128,135,159,177)(129,136,160,178)(130,137,151,179)>;
G:=Group( (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), (1,97,191)(2,98,192)(3,99,193)(4,100,194)(5,91,195)(6,92,196)(7,93,197)(8,94,198)(9,95,199)(10,96,200)(11,179,161)(12,180,162)(13,171,163)(14,172,164)(15,173,165)(16,174,166)(17,175,167)(18,176,168)(19,177,169)(20,178,170)(21,80,135)(22,71,136)(23,72,137)(24,73,138)(25,74,139)(26,75,140)(27,76,131)(28,77,132)(29,78,133)(30,79,134)(31,67,211)(32,68,212)(33,69,213)(34,70,214)(35,61,215)(36,62,216)(37,63,217)(38,64,218)(39,65,219)(40,66,220)(41,105,225)(42,106,226)(43,107,227)(44,108,228)(45,109,229)(46,110,230)(47,101,221)(48,102,222)(49,103,223)(50,104,224)(51,115,209)(52,116,210)(53,117,201)(54,118,202)(55,119,203)(56,120,204)(57,111,205)(58,112,206)(59,113,207)(60,114,208)(81,126,239)(82,127,240)(83,128,231)(84,129,232)(85,130,233)(86,121,234)(87,122,235)(88,123,236)(89,124,237)(90,125,238)(141,160,188)(142,151,189)(143,152,190)(144,153,181)(145,154,182)(146,155,183)(147,156,184)(148,157,185)(149,158,186)(150,159,187), (1,202)(2,203)(3,204)(4,205)(5,206)(6,207)(7,208)(8,209)(9,210)(10,201)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,131)(21,164)(22,165)(23,166)(24,167)(25,168)(26,169)(27,170)(28,161)(29,162)(30,163)(31,50)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(51,198)(52,199)(53,200)(54,191)(55,192)(56,193)(57,194)(58,195)(59,196)(60,197)(61,228)(62,229)(63,230)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,173)(72,174)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,171)(80,172)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,151)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,111)(101,218)(102,219)(103,220)(104,211)(105,212)(106,213)(107,214)(108,215)(109,216)(110,217)(121,148)(122,149)(123,150)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(181,240)(182,231)(183,232)(184,233)(185,234)(186,235)(187,236)(188,237)(189,238)(190,239), (1,237,207,183)(2,238,208,184)(3,239,209,185)(4,240,210,186)(5,231,201,187)(6,232,202,188)(7,233,203,189)(8,234,204,190)(9,235,205,181)(10,236,206,182)(11,218,72,42)(12,219,73,43)(13,220,74,44)(14,211,75,45)(15,212,76,46)(16,213,77,47)(17,214,78,48)(18,215,79,49)(19,216,80,50)(20,217,71,41)(21,224,169,62)(22,225,170,63)(23,226,161,64)(24,227,162,65)(25,228,163,66)(26,229,164,67)(27,230,165,68)(28,221,166,69)(29,222,167,70)(30,223,168,61)(31,140,109,172)(32,131,110,173)(33,132,101,174)(34,133,102,175)(35,134,103,176)(36,135,104,177)(37,136,105,178)(38,137,106,179)(39,138,107,180)(40,139,108,171)(51,148,99,81)(52,149,100,82)(53,150,91,83)(54,141,92,84)(55,142,93,85)(56,143,94,86)(57,144,95,87)(58,145,96,88)(59,146,97,89)(60,147,98,90)(111,153,199,122)(112,154,200,123)(113,155,191,124)(114,156,192,125)(115,157,193,126)(116,158,194,127)(117,159,195,128)(118,160,196,129)(119,151,197,130)(120,152,198,121), (1,230,207,68)(2,221,208,69)(3,222,209,70)(4,223,210,61)(5,224,201,62)(6,225,202,63)(7,226,203,64)(8,227,204,65)(9,228,205,66)(10,229,206,67)(11,85,72,142)(12,86,73,143)(13,87,74,144)(14,88,75,145)(15,89,76,146)(16,90,77,147)(17,81,78,148)(18,82,79,149)(19,83,80,150)(20,84,71,141)(21,187,169,231)(22,188,170,232)(23,189,161,233)(24,190,162,234)(25,181,163,235)(26,182,164,236)(27,183,165,237)(28,184,166,238)(29,185,167,239)(30,186,168,240)(31,200,109,112)(32,191,110,113)(33,192,101,114)(34,193,102,115)(35,194,103,116)(36,195,104,117)(37,196,105,118)(38,197,106,119)(39,198,107,120)(40,199,108,111)(41,54,217,92)(42,55,218,93)(43,56,219,94)(44,57,220,95)(45,58,211,96)(46,59,212,97)(47,60,213,98)(48,51,214,99)(49,52,215,100)(50,53,216,91)(121,138,152,180)(122,139,153,171)(123,140,154,172)(124,131,155,173)(125,132,156,174)(126,133,157,175)(127,134,158,176)(128,135,159,177)(129,136,160,178)(130,137,151,179) );
G=PermutationGroup([[(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)], [(1,97,191),(2,98,192),(3,99,193),(4,100,194),(5,91,195),(6,92,196),(7,93,197),(8,94,198),(9,95,199),(10,96,200),(11,179,161),(12,180,162),(13,171,163),(14,172,164),(15,173,165),(16,174,166),(17,175,167),(18,176,168),(19,177,169),(20,178,170),(21,80,135),(22,71,136),(23,72,137),(24,73,138),(25,74,139),(26,75,140),(27,76,131),(28,77,132),(29,78,133),(30,79,134),(31,67,211),(32,68,212),(33,69,213),(34,70,214),(35,61,215),(36,62,216),(37,63,217),(38,64,218),(39,65,219),(40,66,220),(41,105,225),(42,106,226),(43,107,227),(44,108,228),(45,109,229),(46,110,230),(47,101,221),(48,102,222),(49,103,223),(50,104,224),(51,115,209),(52,116,210),(53,117,201),(54,118,202),(55,119,203),(56,120,204),(57,111,205),(58,112,206),(59,113,207),(60,114,208),(81,126,239),(82,127,240),(83,128,231),(84,129,232),(85,130,233),(86,121,234),(87,122,235),(88,123,236),(89,124,237),(90,125,238),(141,160,188),(142,151,189),(143,152,190),(144,153,181),(145,154,182),(146,155,183),(147,156,184),(148,157,185),(149,158,186),(150,159,187)], [(1,202),(2,203),(3,204),(4,205),(5,206),(6,207),(7,208),(8,209),(9,210),(10,201),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,131),(21,164),(22,165),(23,166),(24,167),(25,168),(26,169),(27,170),(28,161),(29,162),(30,163),(31,50),(32,41),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,49),(51,198),(52,199),(53,200),(54,191),(55,192),(56,193),(57,194),(58,195),(59,196),(60,197),(61,228),(62,229),(63,230),(64,221),(65,222),(66,223),(67,224),(68,225),(69,226),(70,227),(71,173),(72,174),(73,175),(74,176),(75,177),(76,178),(77,179),(78,180),(79,171),(80,172),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,151),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(97,118),(98,119),(99,120),(100,111),(101,218),(102,219),(103,220),(104,211),(105,212),(106,213),(107,214),(108,215),(109,216),(110,217),(121,148),(122,149),(123,150),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(181,240),(182,231),(183,232),(184,233),(185,234),(186,235),(187,236),(188,237),(189,238),(190,239)], [(1,237,207,183),(2,238,208,184),(3,239,209,185),(4,240,210,186),(5,231,201,187),(6,232,202,188),(7,233,203,189),(8,234,204,190),(9,235,205,181),(10,236,206,182),(11,218,72,42),(12,219,73,43),(13,220,74,44),(14,211,75,45),(15,212,76,46),(16,213,77,47),(17,214,78,48),(18,215,79,49),(19,216,80,50),(20,217,71,41),(21,224,169,62),(22,225,170,63),(23,226,161,64),(24,227,162,65),(25,228,163,66),(26,229,164,67),(27,230,165,68),(28,221,166,69),(29,222,167,70),(30,223,168,61),(31,140,109,172),(32,131,110,173),(33,132,101,174),(34,133,102,175),(35,134,103,176),(36,135,104,177),(37,136,105,178),(38,137,106,179),(39,138,107,180),(40,139,108,171),(51,148,99,81),(52,149,100,82),(53,150,91,83),(54,141,92,84),(55,142,93,85),(56,143,94,86),(57,144,95,87),(58,145,96,88),(59,146,97,89),(60,147,98,90),(111,153,199,122),(112,154,200,123),(113,155,191,124),(114,156,192,125),(115,157,193,126),(116,158,194,127),(117,159,195,128),(118,160,196,129),(119,151,197,130),(120,152,198,121)], [(1,230,207,68),(2,221,208,69),(3,222,209,70),(4,223,210,61),(5,224,201,62),(6,225,202,63),(7,226,203,64),(8,227,204,65),(9,228,205,66),(10,229,206,67),(11,85,72,142),(12,86,73,143),(13,87,74,144),(14,88,75,145),(15,89,76,146),(16,90,77,147),(17,81,78,148),(18,82,79,149),(19,83,80,150),(20,84,71,141),(21,187,169,231),(22,188,170,232),(23,189,161,233),(24,190,162,234),(25,181,163,235),(26,182,164,236),(27,183,165,237),(28,184,166,238),(29,185,167,239),(30,186,168,240),(31,200,109,112),(32,191,110,113),(33,192,101,114),(34,193,102,115),(35,194,103,116),(36,195,104,117),(37,196,105,118),(38,197,106,119),(39,198,107,120),(40,199,108,111),(41,54,217,92),(42,55,218,93),(43,56,219,94),(44,57,220,95),(45,58,211,96),(46,59,212,97),(47,60,213,98),(48,51,214,99),(49,52,215,100),(50,53,216,91),(121,138,152,180),(122,139,153,171),(123,140,154,172),(124,131,155,173),(125,132,156,174),(126,133,157,175),(127,134,158,176),(128,135,159,177),(129,136,160,178),(130,137,151,179)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 20Y | ··· | 20AV | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | Q8 | D6 | D6 | C5×S3 | C5×Q8 | S3×C10 | S3×C10 | S3×Q8 | C5×S3×Q8 |
| kernel | S3×Q8×C10 | C10×Dic6 | S3×C2×C20 | C5×S3×Q8 | Q8×C30 | C2×S3×Q8 | C2×Dic6 | S3×C2×C4 | S3×Q8 | C6×Q8 | Q8×C10 | S3×C10 | C2×C20 | C5×Q8 | C2×Q8 | D6 | C2×C4 | Q8 | C10 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 4 | 12 | 12 | 32 | 4 | 1 | 4 | 3 | 4 | 4 | 16 | 12 | 16 | 2 | 8 |
Matrix representation of S3×Q8×C10 ►in GL4(𝔽61) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 60 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 45 | 29 |
| 0 | 0 | 29 | 16 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
G:=sub<GL(4,GF(61))| [3,0,0,0,0,3,0,0,0,0,58,0,0,0,0,58],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,45,29,0,0,29,16],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0] >;
S3×Q8×C10 in GAP, Magma, Sage, TeX
S_3\times Q_8\times C_{10} % in TeX
G:=Group("S3xQ8xC10"); // GroupNames label
G:=SmallGroup(480,1157);
// by ID
G=gap.SmallGroup(480,1157);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,436,633,304,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^3=c^2=d^4=1,e^2=d^2,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