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## G = C10×D4.S3order 480 = 25·3·5

### Direct product of C10 and D4.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C10×D4.S3
 Chief series C1 — C3 — C6 — C12 — C60 — C5×Dic6 — C10×Dic6 — C10×D4.S3
 Lower central C3 — C6 — C12 — C10×D4.S3
 Upper central C1 — C2×C10 — C2×C20 — D4×C10

Generators and relations for C10×D4.S3
G = < a,b,c,d,e | a10=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 292 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×3], C23, C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C30, C30 [×2], C30 [×2], C2×SD16, C40 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×3], C22×C10, C2×C3⋊C8, D4.S3 [×4], C2×Dic6, C6×D4, C5×Dic3 [×2], C60 [×2], C2×C30, C2×C30 [×4], C2×C40, C5×SD16 [×4], D4×C10, Q8×C10, C2×D4.S3, C5×C3⋊C8 [×2], C5×Dic6 [×2], C5×Dic6, C10×Dic3, C2×C60, D4×C15 [×2], D4×C15, C22×C30, C10×SD16, C10×C3⋊C8, C5×D4.S3 [×4], C10×Dic6, D4×C30, C10×D4.S3
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], SD16 [×2], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C2×SD16, C5×D4 [×2], C22×C10, D4.S3 [×2], C2×C3⋊D4, S3×C10 [×3], C5×SD16 [×2], D4×C10, C2×D4.S3, C5×C3⋊D4 [×2], S3×C2×C10, C10×SD16, C5×D4.S3 [×2], C10×C3⋊D4, C10×D4.S3

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

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 30A ··· 30L 30M ··· 30AB 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 5 5 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 4 4 2 2 2 12 12 1 1 1 1 2 2 2 4 4 4 4 6 6 6 6 1 ··· 1 4 ··· 4 4 4 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 D4 D4 D6 D6 SD16 C3⋊D4 C3⋊D4 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×SD16 C5×C3⋊D4 C5×C3⋊D4 D4.S3 C5×D4.S3 kernel C10×D4.S3 C10×C3⋊C8 C5×D4.S3 C10×Dic6 D4×C30 C2×D4.S3 C2×C3⋊C8 D4.S3 C2×Dic6 C6×D4 D4×C10 C60 C2×C30 C2×C20 C5×D4 C30 C20 C2×C10 C2×D4 C12 C2×C6 C2×C4 D4 C6 C4 C22 C10 C2 # reps 1 1 4 1 1 4 4 16 4 4 1 1 1 1 2 4 2 2 4 4 4 4 8 16 8 8 2 8

Matrix representation of C10×D4.S3 in GL4(𝔽241) generated by

 205 0 0 0 0 205 0 0 0 0 36 0 0 0 0 36
,
 0 1 0 0 240 0 0 0 0 0 240 0 0 0 0 240
,
 0 1 0 0 1 0 0 0 0 0 240 0 0 0 29 1
,
 1 0 0 0 0 1 0 0 0 0 15 0 0 0 153 225
,
 222 19 0 0 19 19 0 0 0 0 38 227 0 0 17 203
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,36,0,0,0,0,36],[0,240,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[0,1,0,0,1,0,0,0,0,0,240,29,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,15,153,0,0,0,225],[222,19,0,0,19,19,0,0,0,0,38,17,0,0,227,203] >;

C10×D4.S3 in GAP, Magma, Sage, TeX

C_{10}\times D_4.S_3
% in TeX

G:=Group("C10xD4.S3");
// GroupNames label

G:=SmallGroup(480,812);
// by ID

G=gap.SmallGroup(480,812);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,926,4204,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽