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, Dic3.5(C22×C10), (C5×Dic3).41C23, (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
Subgroups: 548 in 312 conjugacy classes, 194 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C4 [×6], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C2×C4 [×3], C2×C4 [×15], Q8 [×4], Q8 [×12], C23, C10, C10 [×2], C10 [×4], Dic3 [×6], C12 [×6], D6 [×6], C2×C6, C15, C22×C4 [×3], C2×Q8, C2×Q8 [×11], C20 [×6], C20 [×6], C2×C10, C2×C10 [×6], Dic6 [×12], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×3], C3×Q8 [×4], C22×S3, C5×S3 [×4], C30, C30 [×2], C22×Q8, C2×C20 [×3], C2×C20 [×15], C5×Q8 [×4], C5×Q8 [×12], C22×C10, C2×Dic6 [×3], S3×C2×C4 [×3], S3×Q8 [×8], C6×Q8, C5×Dic3 [×6], C60 [×6], S3×C10 [×6], C2×C30, C22×C20 [×3], Q8×C10, Q8×C10 [×11], C2×S3×Q8, C5×Dic6 [×12], S3×C20 [×12], C10×Dic3 [×3], C2×C60 [×3], Q8×C15 [×4], S3×C2×C10, Q8×C2×C10, C10×Dic6 [×3], S3×C2×C20 [×3], C5×S3×Q8 [×8], Q8×C30, S3×Q8×C10
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, Q8 [×4], C23 [×15], C10 [×15], D6 [×7], C2×Q8 [×6], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C22×Q8, C5×Q8 [×4], C22×C10 [×15], S3×Q8 [×2], S3×C23, S3×C10 [×7], Q8×C10 [×6], C23×C10, C2×S3×Q8, S3×C2×C10 [×7], Q8×C2×C10, C5×S3×Q8 [×2], S3×C22×C10, S3×Q8×C10
Generators and relations
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 >
►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)
(1 217 39)(2 218 40)(3 219 31)(4 220 32)(5 211 33)(6 212 34)(7 213 35)(8 214 36)(9 215 37)(10 216 38)(11 236 196)(12 237 197)(13 238 198)(14 239 199)(15 240 200)(16 231 191)(17 232 192)(18 233 193)(19 234 194)(20 235 195)(21 60 135)(22 51 136)(23 52 137)(24 53 138)(25 54 139)(26 55 140)(27 56 131)(28 57 132)(29 58 133)(30 59 134)(41 99 181)(42 100 182)(43 91 183)(44 92 184)(45 93 185)(46 94 186)(47 95 187)(48 96 188)(49 97 189)(50 98 190)(61 89 148)(62 90 149)(63 81 150)(64 82 141)(65 83 142)(66 84 143)(67 85 144)(68 86 145)(69 87 146)(70 88 147)(71 130 156)(72 121 157)(73 122 158)(74 123 159)(75 124 160)(76 125 151)(77 126 152)(78 127 153)(79 128 154)(80 129 155)(101 221 175)(102 222 176)(103 223 177)(104 224 178)(105 225 179)(106 226 180)(107 227 171)(108 228 172)(109 229 173)(110 230 174)(111 205 165)(112 206 166)(113 207 167)(114 208 168)(115 209 169)(116 210 170)(117 201 161)(118 202 162)(119 203 163)(120 204 164)
(1 202)(2 203)(3 204)(4 205)(5 206)(6 207)(7 208)(8 209)(9 210)(10 201)(11 150)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 71)(30 72)(31 164)(32 165)(33 166)(34 167)(35 168)(36 169)(37 170)(38 161)(39 162)(40 163)(41 222)(42 223)(43 224)(44 225)(45 226)(46 227)(47 228)(48 229)(49 230)(50 221)(51 159)(52 160)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 194)(62 195)(63 196)(64 197)(65 198)(66 199)(67 200)(68 191)(69 192)(70 193)(81 236)(82 237)(83 238)(84 239)(85 240)(86 231)(87 232)(88 233)(89 234)(90 235)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 101)(99 102)(100 103)(111 220)(112 211)(113 212)(114 213)(115 214)(116 215)(117 216)(118 217)(119 218)(120 219)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(127 140)(128 131)(129 132)(130 133)(171 186)(172 187)(173 188)(174 189)(175 190)(176 181)(177 182)(178 183)(179 184)(180 185)
(1 237 207 87)(2 238 208 88)(3 239 209 89)(4 240 210 90)(5 231 201 81)(6 232 202 82)(7 233 203 83)(8 234 204 84)(9 235 205 85)(10 236 206 86)(11 112 68 38)(12 113 69 39)(13 114 70 40)(14 115 61 31)(15 116 62 32)(16 117 63 33)(17 118 64 34)(18 119 65 35)(19 120 66 36)(20 111 67 37)(21 224 78 48)(22 225 79 49)(23 226 80 50)(24 227 71 41)(25 228 72 42)(26 229 73 43)(27 230 74 44)(28 221 75 45)(29 222 76 46)(30 223 77 47)(51 179 128 97)(52 180 129 98)(53 171 130 99)(54 172 121 100)(55 173 122 91)(56 174 123 92)(57 175 124 93)(58 176 125 94)(59 177 126 95)(60 178 127 96)(101 160 185 132)(102 151 186 133)(103 152 187 134)(104 153 188 135)(105 154 189 136)(106 155 190 137)(107 156 181 138)(108 157 182 139)(109 158 183 140)(110 159 184 131)(141 212 192 162)(142 213 193 163)(143 214 194 164)(144 215 195 165)(145 216 196 166)(146 217 197 167)(147 218 198 168)(148 219 199 169)(149 220 200 170)(150 211 191 161)
(1 230 207 44)(2 221 208 45)(3 222 209 46)(4 223 210 47)(5 224 201 48)(6 225 202 49)(7 226 203 50)(8 227 204 41)(9 228 205 42)(10 229 206 43)(11 140 68 158)(12 131 69 159)(13 132 70 160)(14 133 61 151)(15 134 62 152)(16 135 63 153)(17 136 64 154)(18 137 65 155)(19 138 66 156)(20 139 67 157)(21 81 78 231)(22 82 79 232)(23 83 80 233)(24 84 71 234)(25 85 72 235)(26 86 73 236)(27 87 74 237)(28 88 75 238)(29 89 76 239)(30 90 77 240)(31 102 115 186)(32 103 116 187)(33 104 117 188)(34 105 118 189)(35 106 119 190)(36 107 120 181)(37 108 111 182)(38 109 112 183)(39 110 113 184)(40 101 114 185)(51 141 128 192)(52 142 129 193)(53 143 130 194)(54 144 121 195)(55 145 122 196)(56 146 123 197)(57 147 124 198)(58 148 125 199)(59 149 126 200)(60 150 127 191)(91 216 173 166)(92 217 174 167)(93 218 175 168)(94 219 176 169)(95 220 177 170)(96 211 178 161)(97 212 179 162)(98 213 180 163)(99 214 171 164)(100 215 172 165)
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), (1,217,39)(2,218,40)(3,219,31)(4,220,32)(5,211,33)(6,212,34)(7,213,35)(8,214,36)(9,215,37)(10,216,38)(11,236,196)(12,237,197)(13,238,198)(14,239,199)(15,240,200)(16,231,191)(17,232,192)(18,233,193)(19,234,194)(20,235,195)(21,60,135)(22,51,136)(23,52,137)(24,53,138)(25,54,139)(26,55,140)(27,56,131)(28,57,132)(29,58,133)(30,59,134)(41,99,181)(42,100,182)(43,91,183)(44,92,184)(45,93,185)(46,94,186)(47,95,187)(48,96,188)(49,97,189)(50,98,190)(61,89,148)(62,90,149)(63,81,150)(64,82,141)(65,83,142)(66,84,143)(67,85,144)(68,86,145)(69,87,146)(70,88,147)(71,130,156)(72,121,157)(73,122,158)(74,123,159)(75,124,160)(76,125,151)(77,126,152)(78,127,153)(79,128,154)(80,129,155)(101,221,175)(102,222,176)(103,223,177)(104,224,178)(105,225,179)(106,226,180)(107,227,171)(108,228,172)(109,229,173)(110,230,174)(111,205,165)(112,206,166)(113,207,167)(114,208,168)(115,209,169)(116,210,170)(117,201,161)(118,202,162)(119,203,163)(120,204,164), (1,202)(2,203)(3,204)(4,205)(5,206)(6,207)(7,208)(8,209)(9,210)(10,201)(11,150)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,71)(30,72)(31,164)(32,165)(33,166)(34,167)(35,168)(36,169)(37,170)(38,161)(39,162)(40,163)(41,222)(42,223)(43,224)(44,225)(45,226)(46,227)(47,228)(48,229)(49,230)(50,221)(51,159)(52,160)(53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,157)(60,158)(61,194)(62,195)(63,196)(64,197)(65,198)(66,199)(67,200)(68,191)(69,192)(70,193)(81,236)(82,237)(83,238)(84,239)(85,240)(86,231)(87,232)(88,233)(89,234)(90,235)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,101)(99,102)(100,103)(111,220)(112,211)(113,212)(114,213)(115,214)(116,215)(117,216)(118,217)(119,218)(120,219)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,131)(129,132)(130,133)(171,186)(172,187)(173,188)(174,189)(175,190)(176,181)(177,182)(178,183)(179,184)(180,185), (1,237,207,87)(2,238,208,88)(3,239,209,89)(4,240,210,90)(5,231,201,81)(6,232,202,82)(7,233,203,83)(8,234,204,84)(9,235,205,85)(10,236,206,86)(11,112,68,38)(12,113,69,39)(13,114,70,40)(14,115,61,31)(15,116,62,32)(16,117,63,33)(17,118,64,34)(18,119,65,35)(19,120,66,36)(20,111,67,37)(21,224,78,48)(22,225,79,49)(23,226,80,50)(24,227,71,41)(25,228,72,42)(26,229,73,43)(27,230,74,44)(28,221,75,45)(29,222,76,46)(30,223,77,47)(51,179,128,97)(52,180,129,98)(53,171,130,99)(54,172,121,100)(55,173,122,91)(56,174,123,92)(57,175,124,93)(58,176,125,94)(59,177,126,95)(60,178,127,96)(101,160,185,132)(102,151,186,133)(103,152,187,134)(104,153,188,135)(105,154,189,136)(106,155,190,137)(107,156,181,138)(108,157,182,139)(109,158,183,140)(110,159,184,131)(141,212,192,162)(142,213,193,163)(143,214,194,164)(144,215,195,165)(145,216,196,166)(146,217,197,167)(147,218,198,168)(148,219,199,169)(149,220,200,170)(150,211,191,161), (1,230,207,44)(2,221,208,45)(3,222,209,46)(4,223,210,47)(5,224,201,48)(6,225,202,49)(7,226,203,50)(8,227,204,41)(9,228,205,42)(10,229,206,43)(11,140,68,158)(12,131,69,159)(13,132,70,160)(14,133,61,151)(15,134,62,152)(16,135,63,153)(17,136,64,154)(18,137,65,155)(19,138,66,156)(20,139,67,157)(21,81,78,231)(22,82,79,232)(23,83,80,233)(24,84,71,234)(25,85,72,235)(26,86,73,236)(27,87,74,237)(28,88,75,238)(29,89,76,239)(30,90,77,240)(31,102,115,186)(32,103,116,187)(33,104,117,188)(34,105,118,189)(35,106,119,190)(36,107,120,181)(37,108,111,182)(38,109,112,183)(39,110,113,184)(40,101,114,185)(51,141,128,192)(52,142,129,193)(53,143,130,194)(54,144,121,195)(55,145,122,196)(56,146,123,197)(57,147,124,198)(58,148,125,199)(59,149,126,200)(60,150,127,191)(91,216,173,166)(92,217,174,167)(93,218,175,168)(94,219,176,169)(95,220,177,170)(96,211,178,161)(97,212,179,162)(98,213,180,163)(99,214,171,164)(100,215,172,165)>;
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), (1,217,39)(2,218,40)(3,219,31)(4,220,32)(5,211,33)(6,212,34)(7,213,35)(8,214,36)(9,215,37)(10,216,38)(11,236,196)(12,237,197)(13,238,198)(14,239,199)(15,240,200)(16,231,191)(17,232,192)(18,233,193)(19,234,194)(20,235,195)(21,60,135)(22,51,136)(23,52,137)(24,53,138)(25,54,139)(26,55,140)(27,56,131)(28,57,132)(29,58,133)(30,59,134)(41,99,181)(42,100,182)(43,91,183)(44,92,184)(45,93,185)(46,94,186)(47,95,187)(48,96,188)(49,97,189)(50,98,190)(61,89,148)(62,90,149)(63,81,150)(64,82,141)(65,83,142)(66,84,143)(67,85,144)(68,86,145)(69,87,146)(70,88,147)(71,130,156)(72,121,157)(73,122,158)(74,123,159)(75,124,160)(76,125,151)(77,126,152)(78,127,153)(79,128,154)(80,129,155)(101,221,175)(102,222,176)(103,223,177)(104,224,178)(105,225,179)(106,226,180)(107,227,171)(108,228,172)(109,229,173)(110,230,174)(111,205,165)(112,206,166)(113,207,167)(114,208,168)(115,209,169)(116,210,170)(117,201,161)(118,202,162)(119,203,163)(120,204,164), (1,202)(2,203)(3,204)(4,205)(5,206)(6,207)(7,208)(8,209)(9,210)(10,201)(11,150)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,71)(30,72)(31,164)(32,165)(33,166)(34,167)(35,168)(36,169)(37,170)(38,161)(39,162)(40,163)(41,222)(42,223)(43,224)(44,225)(45,226)(46,227)(47,228)(48,229)(49,230)(50,221)(51,159)(52,160)(53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,157)(60,158)(61,194)(62,195)(63,196)(64,197)(65,198)(66,199)(67,200)(68,191)(69,192)(70,193)(81,236)(82,237)(83,238)(84,239)(85,240)(86,231)(87,232)(88,233)(89,234)(90,235)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,101)(99,102)(100,103)(111,220)(112,211)(113,212)(114,213)(115,214)(116,215)(117,216)(118,217)(119,218)(120,219)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,131)(129,132)(130,133)(171,186)(172,187)(173,188)(174,189)(175,190)(176,181)(177,182)(178,183)(179,184)(180,185), (1,237,207,87)(2,238,208,88)(3,239,209,89)(4,240,210,90)(5,231,201,81)(6,232,202,82)(7,233,203,83)(8,234,204,84)(9,235,205,85)(10,236,206,86)(11,112,68,38)(12,113,69,39)(13,114,70,40)(14,115,61,31)(15,116,62,32)(16,117,63,33)(17,118,64,34)(18,119,65,35)(19,120,66,36)(20,111,67,37)(21,224,78,48)(22,225,79,49)(23,226,80,50)(24,227,71,41)(25,228,72,42)(26,229,73,43)(27,230,74,44)(28,221,75,45)(29,222,76,46)(30,223,77,47)(51,179,128,97)(52,180,129,98)(53,171,130,99)(54,172,121,100)(55,173,122,91)(56,174,123,92)(57,175,124,93)(58,176,125,94)(59,177,126,95)(60,178,127,96)(101,160,185,132)(102,151,186,133)(103,152,187,134)(104,153,188,135)(105,154,189,136)(106,155,190,137)(107,156,181,138)(108,157,182,139)(109,158,183,140)(110,159,184,131)(141,212,192,162)(142,213,193,163)(143,214,194,164)(144,215,195,165)(145,216,196,166)(146,217,197,167)(147,218,198,168)(148,219,199,169)(149,220,200,170)(150,211,191,161), (1,230,207,44)(2,221,208,45)(3,222,209,46)(4,223,210,47)(5,224,201,48)(6,225,202,49)(7,226,203,50)(8,227,204,41)(9,228,205,42)(10,229,206,43)(11,140,68,158)(12,131,69,159)(13,132,70,160)(14,133,61,151)(15,134,62,152)(16,135,63,153)(17,136,64,154)(18,137,65,155)(19,138,66,156)(20,139,67,157)(21,81,78,231)(22,82,79,232)(23,83,80,233)(24,84,71,234)(25,85,72,235)(26,86,73,236)(27,87,74,237)(28,88,75,238)(29,89,76,239)(30,90,77,240)(31,102,115,186)(32,103,116,187)(33,104,117,188)(34,105,118,189)(35,106,119,190)(36,107,120,181)(37,108,111,182)(38,109,112,183)(39,110,113,184)(40,101,114,185)(51,141,128,192)(52,142,129,193)(53,143,130,194)(54,144,121,195)(55,145,122,196)(56,146,123,197)(57,147,124,198)(58,148,125,199)(59,149,126,200)(60,150,127,191)(91,216,173,166)(92,217,174,167)(93,218,175,168)(94,219,176,169)(95,220,177,170)(96,211,178,161)(97,212,179,162)(98,213,180,163)(99,214,171,164)(100,215,172,165) );
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)], [(1,217,39),(2,218,40),(3,219,31),(4,220,32),(5,211,33),(6,212,34),(7,213,35),(8,214,36),(9,215,37),(10,216,38),(11,236,196),(12,237,197),(13,238,198),(14,239,199),(15,240,200),(16,231,191),(17,232,192),(18,233,193),(19,234,194),(20,235,195),(21,60,135),(22,51,136),(23,52,137),(24,53,138),(25,54,139),(26,55,140),(27,56,131),(28,57,132),(29,58,133),(30,59,134),(41,99,181),(42,100,182),(43,91,183),(44,92,184),(45,93,185),(46,94,186),(47,95,187),(48,96,188),(49,97,189),(50,98,190),(61,89,148),(62,90,149),(63,81,150),(64,82,141),(65,83,142),(66,84,143),(67,85,144),(68,86,145),(69,87,146),(70,88,147),(71,130,156),(72,121,157),(73,122,158),(74,123,159),(75,124,160),(76,125,151),(77,126,152),(78,127,153),(79,128,154),(80,129,155),(101,221,175),(102,222,176),(103,223,177),(104,224,178),(105,225,179),(106,226,180),(107,227,171),(108,228,172),(109,229,173),(110,230,174),(111,205,165),(112,206,166),(113,207,167),(114,208,168),(115,209,169),(116,210,170),(117,201,161),(118,202,162),(119,203,163),(120,204,164)], [(1,202),(2,203),(3,204),(4,205),(5,206),(6,207),(7,208),(8,209),(9,210),(10,201),(11,150),(12,141),(13,142),(14,143),(15,144),(16,145),(17,146),(18,147),(19,148),(20,149),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,71),(30,72),(31,164),(32,165),(33,166),(34,167),(35,168),(36,169),(37,170),(38,161),(39,162),(40,163),(41,222),(42,223),(43,224),(44,225),(45,226),(46,227),(47,228),(48,229),(49,230),(50,221),(51,159),(52,160),(53,151),(54,152),(55,153),(56,154),(57,155),(58,156),(59,157),(60,158),(61,194),(62,195),(63,196),(64,197),(65,198),(66,199),(67,200),(68,191),(69,192),(70,193),(81,236),(82,237),(83,238),(84,239),(85,240),(86,231),(87,232),(88,233),(89,234),(90,235),(91,104),(92,105),(93,106),(94,107),(95,108),(96,109),(97,110),(98,101),(99,102),(100,103),(111,220),(112,211),(113,212),(114,213),(115,214),(116,215),(117,216),(118,217),(119,218),(120,219),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(127,140),(128,131),(129,132),(130,133),(171,186),(172,187),(173,188),(174,189),(175,190),(176,181),(177,182),(178,183),(179,184),(180,185)], [(1,237,207,87),(2,238,208,88),(3,239,209,89),(4,240,210,90),(5,231,201,81),(6,232,202,82),(7,233,203,83),(8,234,204,84),(9,235,205,85),(10,236,206,86),(11,112,68,38),(12,113,69,39),(13,114,70,40),(14,115,61,31),(15,116,62,32),(16,117,63,33),(17,118,64,34),(18,119,65,35),(19,120,66,36),(20,111,67,37),(21,224,78,48),(22,225,79,49),(23,226,80,50),(24,227,71,41),(25,228,72,42),(26,229,73,43),(27,230,74,44),(28,221,75,45),(29,222,76,46),(30,223,77,47),(51,179,128,97),(52,180,129,98),(53,171,130,99),(54,172,121,100),(55,173,122,91),(56,174,123,92),(57,175,124,93),(58,176,125,94),(59,177,126,95),(60,178,127,96),(101,160,185,132),(102,151,186,133),(103,152,187,134),(104,153,188,135),(105,154,189,136),(106,155,190,137),(107,156,181,138),(108,157,182,139),(109,158,183,140),(110,159,184,131),(141,212,192,162),(142,213,193,163),(143,214,194,164),(144,215,195,165),(145,216,196,166),(146,217,197,167),(147,218,198,168),(148,219,199,169),(149,220,200,170),(150,211,191,161)], [(1,230,207,44),(2,221,208,45),(3,222,209,46),(4,223,210,47),(5,224,201,48),(6,225,202,49),(7,226,203,50),(8,227,204,41),(9,228,205,42),(10,229,206,43),(11,140,68,158),(12,131,69,159),(13,132,70,160),(14,133,61,151),(15,134,62,152),(16,135,63,153),(17,136,64,154),(18,137,65,155),(19,138,66,156),(20,139,67,157),(21,81,78,231),(22,82,79,232),(23,83,80,233),(24,84,71,234),(25,85,72,235),(26,86,73,236),(27,87,74,237),(28,88,75,238),(29,89,76,239),(30,90,77,240),(31,102,115,186),(32,103,116,187),(33,104,117,188),(34,105,118,189),(35,106,119,190),(36,107,120,181),(37,108,111,182),(38,109,112,183),(39,110,113,184),(40,101,114,185),(51,141,128,192),(52,142,129,193),(53,143,130,194),(54,144,121,195),(55,145,122,196),(56,146,123,197),(57,147,124,198),(58,148,125,199),(59,149,126,200),(60,150,127,191),(91,216,173,166),(92,217,174,167),(93,218,175,168),(94,219,176,169),(95,220,177,170),(96,211,178,161),(97,212,179,162),(98,213,180,163),(99,214,171,164),(100,215,172,165)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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