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## G = S3×C10.D4order 480 = 25·3·5

### Direct product of S3 and C10.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×C10.D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×S3×Dic5 — S3×C10.D4
 Lower central C15 — C30 — S3×C10.D4
 Upper central C1 — C22 — C2×C4

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 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×13], C23, C10 [×3], C10 [×4], Dic3 [×4], C12 [×4], D6 [×6], C2×C6, C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], C4×S3 [×8], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×4], C30 [×3], C2×C4⋊C4, C2×Dic5 [×2], C2×Dic5 [×8], C2×C20, C2×C20 [×3], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, S3×C10 [×6], C2×C30, C10.D4, C10.D4 [×3], C22×Dic5 [×2], C22×C20, S3×C4⋊C4, S3×Dic5 [×4], S3×Dic5 [×2], C6×Dic5 [×2], S3×C20 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C2×C10.D4, C30.Q8, Dic155C4, C3×C10.D4, C30.4Q8, C2×S3×Dic5 [×2], S3×C2×C20, S3×C10.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C4×S3 [×2], C22×S3, C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, S3×C2×C4, S3×D4, S3×Q8, S3×D5, C10.D4 [×4], 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

Smallest permutation representation of S3×C10.D4
On 240 points
Generators in S240
(1 25 43)(2 26 44)(3 27 45)(4 28 46)(5 29 47)(6 30 48)(7 21 49)(8 22 50)(9 23 41)(10 24 42)(11 67 59)(12 68 60)(13 69 51)(14 70 52)(15 61 53)(16 62 54)(17 63 55)(18 64 56)(19 65 57)(20 66 58)(31 237 216)(32 238 217)(33 239 218)(34 240 219)(35 231 220)(36 232 211)(37 233 212)(38 234 213)(39 235 214)(40 236 215)(71 85 103)(72 86 104)(73 87 105)(74 88 106)(75 89 107)(76 90 108)(77 81 109)(78 82 110)(79 83 101)(80 84 102)(91 127 120)(92 128 111)(93 129 112)(94 130 113)(95 121 114)(96 122 115)(97 123 116)(98 124 117)(99 125 118)(100 126 119)(131 147 164)(132 148 165)(133 149 166)(134 150 167)(135 141 168)(136 142 169)(137 143 170)(138 144 161)(139 145 162)(140 146 163)(151 189 172)(152 190 173)(153 181 174)(154 182 175)(155 183 176)(156 184 177)(157 185 178)(158 186 179)(159 187 180)(160 188 171)(191 207 224)(192 208 225)(193 209 226)(194 210 227)(195 201 228)(196 202 229)(197 203 230)(198 204 221)(199 205 222)(200 206 223)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 50)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(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)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 111)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)(87 121)(88 122)(89 123)(90 124)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)(99 109)(100 110)(131 175)(132 176)(133 177)(134 178)(135 179)(136 180)(137 171)(138 172)(139 173)(140 174)(141 186)(142 187)(143 188)(144 189)(145 190)(146 181)(147 182)(148 183)(149 184)(150 185)(151 161)(152 162)(153 163)(154 164)(155 165)(156 166)(157 167)(158 168)(159 169)(160 170)(191 235)(192 236)(193 237)(194 238)(195 239)(196 240)(197 231)(198 232)(199 233)(200 234)(211 221)(212 222)(213 223)(214 224)(215 225)(216 226)(217 227)(218 228)(219 229)(220 230)
(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 57 134)(2 172 58 133)(3 171 59 132)(4 180 60 131)(5 179 51 140)(6 178 52 139)(7 177 53 138)(8 176 54 137)(9 175 55 136)(10 174 56 135)(11 148 27 160)(12 147 28 159)(13 146 29 158)(14 145 30 157)(15 144 21 156)(16 143 22 155)(17 142 23 154)(18 141 24 153)(19 150 25 152)(20 149 26 151)(31 130 221 109)(32 129 222 108)(33 128 223 107)(34 127 224 106)(35 126 225 105)(36 125 226 104)(37 124 227 103)(38 123 228 102)(39 122 229 101)(40 121 230 110)(41 182 63 169)(42 181 64 168)(43 190 65 167)(44 189 66 166)(45 188 67 165)(46 187 68 164)(47 186 69 163)(48 185 70 162)(49 184 61 161)(50 183 62 170)(71 233 117 194)(72 232 118 193)(73 231 119 192)(74 240 120 191)(75 239 111 200)(76 238 112 199)(77 237 113 198)(78 236 114 197)(79 235 115 196)(80 234 116 195)(81 216 94 204)(82 215 95 203)(83 214 96 202)(84 213 97 201)(85 212 98 210)(86 211 99 209)(87 220 100 208)(88 219 91 207)(89 218 92 206)(90 217 93 205)
(1 199 6 194)(2 198 7 193)(3 197 8 192)(4 196 9 191)(5 195 10 200)(11 215 16 220)(12 214 17 219)(13 213 18 218)(14 212 19 217)(15 211 20 216)(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)(41 224 46 229)(42 223 47 228)(43 222 48 227)(44 221 49 226)(45 230 50 225)(51 234 56 239)(52 233 57 238)(53 232 58 237)(54 231 59 236)(55 240 60 235)(71 134 76 139)(72 133 77 138)(73 132 78 137)(74 131 79 136)(75 140 80 135)(81 144 86 149)(82 143 87 148)(83 142 88 147)(84 141 89 146)(85 150 90 145)(91 159 96 154)(92 158 97 153)(93 157 98 152)(94 156 99 151)(95 155 100 160)(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)| (1,25,43)(2,26,44)(3,27,45)(4,28,46)(5,29,47)(6,30,48)(7,21,49)(8,22,50)(9,23,41)(10,24,42)(11,67,59)(12,68,60)(13,69,51)(14,70,52)(15,61,53)(16,62,54)(17,63,55)(18,64,56)(19,65,57)(20,66,58)(31,237,216)(32,238,217)(33,239,218)(34,240,219)(35,231,220)(36,232,211)(37,233,212)(38,234,213)(39,235,214)(40,236,215)(71,85,103)(72,86,104)(73,87,105)(74,88,106)(75,89,107)(76,90,108)(77,81,109)(78,82,110)(79,83,101)(80,84,102)(91,127,120)(92,128,111)(93,129,112)(94,130,113)(95,121,114)(96,122,115)(97,123,116)(98,124,117)(99,125,118)(100,126,119)(131,147,164)(132,148,165)(133,149,166)(134,150,167)(135,141,168)(136,142,169)(137,143,170)(138,144,161)(139,145,162)(140,146,163)(151,189,172)(152,190,173)(153,181,174)(154,182,175)(155,183,176)(156,184,177)(157,185,178)(158,186,179)(159,187,180)(160,188,171)(191,207,224)(192,208,225)(193,209,226)(194,210,227)(195,201,228)(196,202,229)(197,203,230)(198,204,221)(199,205,222)(200,206,223), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,50)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(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)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,111)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,121)(88,122)(89,123)(90,124)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108)(99,109)(100,110)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,171)(138,172)(139,173)(140,174)(141,186)(142,187)(143,188)(144,189)(145,190)(146,181)(147,182)(148,183)(149,184)(150,185)(151,161)(152,162)(153,163)(154,164)(155,165)(156,166)(157,167)(158,168)(159,169)(160,170)(191,235)(192,236)(193,237)(194,238)(195,239)(196,240)(197,231)(198,232)(199,233)(200,234)(211,221)(212,222)(213,223)(214,224)(215,225)(216,226)(217,227)(218,228)(219,229)(220,230), (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), (1,173,57,134)(2,172,58,133)(3,171,59,132)(4,180,60,131)(5,179,51,140)(6,178,52,139)(7,177,53,138)(8,176,54,137)(9,175,55,136)(10,174,56,135)(11,148,27,160)(12,147,28,159)(13,146,29,158)(14,145,30,157)(15,144,21,156)(16,143,22,155)(17,142,23,154)(18,141,24,153)(19,150,25,152)(20,149,26,151)(31,130,221,109)(32,129,222,108)(33,128,223,107)(34,127,224,106)(35,126,225,105)(36,125,226,104)(37,124,227,103)(38,123,228,102)(39,122,229,101)(40,121,230,110)(41,182,63,169)(42,181,64,168)(43,190,65,167)(44,189,66,166)(45,188,67,165)(46,187,68,164)(47,186,69,163)(48,185,70,162)(49,184,61,161)(50,183,62,170)(71,233,117,194)(72,232,118,193)(73,231,119,192)(74,240,120,191)(75,239,111,200)(76,238,112,199)(77,237,113,198)(78,236,114,197)(79,235,115,196)(80,234,116,195)(81,216,94,204)(82,215,95,203)(83,214,96,202)(84,213,97,201)(85,212,98,210)(86,211,99,209)(87,220,100,208)(88,219,91,207)(89,218,92,206)(90,217,93,205), (1,199,6,194)(2,198,7,193)(3,197,8,192)(4,196,9,191)(5,195,10,200)(11,215,16,220)(12,214,17,219)(13,213,18,218)(14,212,19,217)(15,211,20,216)(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)(41,224,46,229)(42,223,47,228)(43,222,48,227)(44,221,49,226)(45,230,50,225)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,134,76,139)(72,133,77,138)(73,132,78,137)(74,131,79,136)(75,140,80,135)(81,144,86,149)(82,143,87,148)(83,142,88,147)(84,141,89,146)(85,150,90,145)(91,159,96,154)(92,158,97,153)(93,157,98,152)(94,156,99,151)(95,155,100,160)(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( (1,25,43)(2,26,44)(3,27,45)(4,28,46)(5,29,47)(6,30,48)(7,21,49)(8,22,50)(9,23,41)(10,24,42)(11,67,59)(12,68,60)(13,69,51)(14,70,52)(15,61,53)(16,62,54)(17,63,55)(18,64,56)(19,65,57)(20,66,58)(31,237,216)(32,238,217)(33,239,218)(34,240,219)(35,231,220)(36,232,211)(37,233,212)(38,234,213)(39,235,214)(40,236,215)(71,85,103)(72,86,104)(73,87,105)(74,88,106)(75,89,107)(76,90,108)(77,81,109)(78,82,110)(79,83,101)(80,84,102)(91,127,120)(92,128,111)(93,129,112)(94,130,113)(95,121,114)(96,122,115)(97,123,116)(98,124,117)(99,125,118)(100,126,119)(131,147,164)(132,148,165)(133,149,166)(134,150,167)(135,141,168)(136,142,169)(137,143,170)(138,144,161)(139,145,162)(140,146,163)(151,189,172)(152,190,173)(153,181,174)(154,182,175)(155,183,176)(156,184,177)(157,185,178)(158,186,179)(159,187,180)(160,188,171)(191,207,224)(192,208,225)(193,209,226)(194,210,227)(195,201,228)(196,202,229)(197,203,230)(198,204,221)(199,205,222)(200,206,223), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,50)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(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)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,111)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,121)(88,122)(89,123)(90,124)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108)(99,109)(100,110)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,171)(138,172)(139,173)(140,174)(141,186)(142,187)(143,188)(144,189)(145,190)(146,181)(147,182)(148,183)(149,184)(150,185)(151,161)(152,162)(153,163)(154,164)(155,165)(156,166)(157,167)(158,168)(159,169)(160,170)(191,235)(192,236)(193,237)(194,238)(195,239)(196,240)(197,231)(198,232)(199,233)(200,234)(211,221)(212,222)(213,223)(214,224)(215,225)(216,226)(217,227)(218,228)(219,229)(220,230), (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), (1,173,57,134)(2,172,58,133)(3,171,59,132)(4,180,60,131)(5,179,51,140)(6,178,52,139)(7,177,53,138)(8,176,54,137)(9,175,55,136)(10,174,56,135)(11,148,27,160)(12,147,28,159)(13,146,29,158)(14,145,30,157)(15,144,21,156)(16,143,22,155)(17,142,23,154)(18,141,24,153)(19,150,25,152)(20,149,26,151)(31,130,221,109)(32,129,222,108)(33,128,223,107)(34,127,224,106)(35,126,225,105)(36,125,226,104)(37,124,227,103)(38,123,228,102)(39,122,229,101)(40,121,230,110)(41,182,63,169)(42,181,64,168)(43,190,65,167)(44,189,66,166)(45,188,67,165)(46,187,68,164)(47,186,69,163)(48,185,70,162)(49,184,61,161)(50,183,62,170)(71,233,117,194)(72,232,118,193)(73,231,119,192)(74,240,120,191)(75,239,111,200)(76,238,112,199)(77,237,113,198)(78,236,114,197)(79,235,115,196)(80,234,116,195)(81,216,94,204)(82,215,95,203)(83,214,96,202)(84,213,97,201)(85,212,98,210)(86,211,99,209)(87,220,100,208)(88,219,91,207)(89,218,92,206)(90,217,93,205), (1,199,6,194)(2,198,7,193)(3,197,8,192)(4,196,9,191)(5,195,10,200)(11,215,16,220)(12,214,17,219)(13,213,18,218)(14,212,19,217)(15,211,20,216)(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)(41,224,46,229)(42,223,47,228)(43,222,48,227)(44,221,49,226)(45,230,50,225)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,134,76,139)(72,133,77,138)(73,132,78,137)(74,131,79,136)(75,140,80,135)(81,144,86,149)(82,143,87,148)(83,142,88,147)(84,141,89,146)(85,150,90,145)(91,159,96,154)(92,158,97,153)(93,157,98,152)(94,156,99,151)(95,155,100,160)(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([(1,25,43),(2,26,44),(3,27,45),(4,28,46),(5,29,47),(6,30,48),(7,21,49),(8,22,50),(9,23,41),(10,24,42),(11,67,59),(12,68,60),(13,69,51),(14,70,52),(15,61,53),(16,62,54),(17,63,55),(18,64,56),(19,65,57),(20,66,58),(31,237,216),(32,238,217),(33,239,218),(34,240,219),(35,231,220),(36,232,211),(37,233,212),(38,234,213),(39,235,214),(40,236,215),(71,85,103),(72,86,104),(73,87,105),(74,88,106),(75,89,107),(76,90,108),(77,81,109),(78,82,110),(79,83,101),(80,84,102),(91,127,120),(92,128,111),(93,129,112),(94,130,113),(95,121,114),(96,122,115),(97,123,116),(98,124,117),(99,125,118),(100,126,119),(131,147,164),(132,148,165),(133,149,166),(134,150,167),(135,141,168),(136,142,169),(137,143,170),(138,144,161),(139,145,162),(140,146,163),(151,189,172),(152,190,173),(153,181,174),(154,182,175),(155,183,176),(156,184,177),(157,185,178),(158,186,179),(159,187,180),(160,188,171),(191,207,224),(192,208,225),(193,209,226),(194,210,227),(195,201,228),(196,202,229),(197,203,230),(198,204,221),(199,205,222),(200,206,223)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,51),(11,50),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(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),(71,112),(72,113),(73,114),(74,115),(75,116),(76,117),(77,118),(78,119),(79,120),(80,111),(81,125),(82,126),(83,127),(84,128),(85,129),(86,130),(87,121),(88,122),(89,123),(90,124),(91,101),(92,102),(93,103),(94,104),(95,105),(96,106),(97,107),(98,108),(99,109),(100,110),(131,175),(132,176),(133,177),(134,178),(135,179),(136,180),(137,171),(138,172),(139,173),(140,174),(141,186),(142,187),(143,188),(144,189),(145,190),(146,181),(147,182),(148,183),(149,184),(150,185),(151,161),(152,162),(153,163),(154,164),(155,165),(156,166),(157,167),(158,168),(159,169),(160,170),(191,235),(192,236),(193,237),(194,238),(195,239),(196,240),(197,231),(198,232),(199,233),(200,234),(211,221),(212,222),(213,223),(214,224),(215,225),(216,226),(217,227),(218,228),(219,229),(220,230)], [(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)], [(1,173,57,134),(2,172,58,133),(3,171,59,132),(4,180,60,131),(5,179,51,140),(6,178,52,139),(7,177,53,138),(8,176,54,137),(9,175,55,136),(10,174,56,135),(11,148,27,160),(12,147,28,159),(13,146,29,158),(14,145,30,157),(15,144,21,156),(16,143,22,155),(17,142,23,154),(18,141,24,153),(19,150,25,152),(20,149,26,151),(31,130,221,109),(32,129,222,108),(33,128,223,107),(34,127,224,106),(35,126,225,105),(36,125,226,104),(37,124,227,103),(38,123,228,102),(39,122,229,101),(40,121,230,110),(41,182,63,169),(42,181,64,168),(43,190,65,167),(44,189,66,166),(45,188,67,165),(46,187,68,164),(47,186,69,163),(48,185,70,162),(49,184,61,161),(50,183,62,170),(71,233,117,194),(72,232,118,193),(73,231,119,192),(74,240,120,191),(75,239,111,200),(76,238,112,199),(77,237,113,198),(78,236,114,197),(79,235,115,196),(80,234,116,195),(81,216,94,204),(82,215,95,203),(83,214,96,202),(84,213,97,201),(85,212,98,210),(86,211,99,209),(87,220,100,208),(88,219,91,207),(89,218,92,206),(90,217,93,205)], [(1,199,6,194),(2,198,7,193),(3,197,8,192),(4,196,9,191),(5,195,10,200),(11,215,16,220),(12,214,17,219),(13,213,18,218),(14,212,19,217),(15,211,20,216),(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),(41,224,46,229),(42,223,47,228),(43,222,48,227),(44,221,49,226),(45,230,50,225),(51,234,56,239),(52,233,57,238),(53,232,58,237),(54,231,59,236),(55,240,60,235),(71,134,76,139),(72,133,77,138),(73,132,78,137),(74,131,79,136),(75,140,80,135),(81,144,86,149),(82,143,87,148),(83,142,88,147),(84,141,89,146),(85,150,90,145),(91,159,96,154),(92,158,97,153),(93,157,98,152),(94,156,99,151),(95,155,100,160),(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

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