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G = S3×Q8×C10order 480 = 25·3·5

Direct product of C10, S3 and Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×Q8×C10, C30.91C24, C60.238C23, C62(Q8×C10), (C6×Q8)⋊5C10, C3010(C2×Q8), (Q8×C30)⋊19C2, C1511(C22×Q8), Dic69(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, C32(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

C1C6 — S3×Q8×C10
C1C3C6C30S3×C10S3×C2×C10S3×C2×C20 — S3×Q8×C10
C3C6 — S3×Q8×C10
C1C2×C10Q8×C10

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 [×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

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

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

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

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

150 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4F4G···4L5A5B5C5D6A6B6C10A···10L10M···10AB12A···12F15A15B15C15D20A···20X20Y···20AV30A···30L60A···60X
order1222222234···44···4555566610···1010···1012···121515151520···2020···2030···3060···60
size1111333322···26···611112221···13···34···422222···26···62···24···4

150 irreducible representations

dim11111111112222222244
type++++++-++-
imageC1C2C2C2C2C5C10C10C10C10S3Q8D6D6C5×S3C5×Q8S3×C10S3×C10S3×Q8C5×S3×Q8
kernelS3×Q8×C10C10×Dic6S3×C2×C20C5×S3×Q8Q8×C30C2×S3×Q8C2×Dic6S3×C2×C4S3×Q8C6×Q8Q8×C10S3×C10C2×C20C5×Q8C2×Q8D6C2×C4Q8C10C2
# reps13381412123241434416121628

Matrix representation of S3×Q8×C10 in GL4(𝔽61) generated by

3000
0300
00580
00058
,
606000
1000
0010
0001
,
60000
1100
00600
00060
,
1000
0100
004529
002916
,
60000
06000
0001
00600
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

׿
×
𝔽