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## G = C5×D8⋊3S3order 480 = 25·3·5

### Direct product of C5 and D8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×D8⋊3S3
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×D4⋊2S3 — C5×D8⋊3S3
 Lower central C3 — C6 — C12 — C5×D8⋊3S3
 Upper central C1 — C10 — C20 — C5×D8

Generators and relations for C5×D83S3
G = < a,b,c,d,e | a5=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 292 in 124 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C5×S3, C30, C30, C4○D8, C40, C40, C2×C20, C5×D4, C5×D4, C5×Q8, S3×C8, Dic12, D4.S3, C3×D8, D42S3, C5×Dic3, C5×Dic3, C60, S3×C10, C2×C30, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, D83S3, C5×C3⋊C8, C120, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, D4×C15, C5×C4○D8, S3×C40, C5×Dic12, C5×D4.S3, C15×D8, C5×D42S3, C5×D83S3
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C4○D8, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, D83S3, S3×C2×C10, C5×C4○D8, C5×S3×D4, C5×D83S3

Smallest permutation representation of C5×D83S3
On 240 points
Generators in S240
(1 181 74 57 125)(2 182 75 58 126)(3 183 76 59 127)(4 184 77 60 128)(5 177 78 61 121)(6 178 79 62 122)(7 179 80 63 123)(8 180 73 64 124)(9 139 230 69 84)(10 140 231 70 85)(11 141 232 71 86)(12 142 225 72 87)(13 143 226 65 88)(14 144 227 66 81)(15 137 228 67 82)(16 138 229 68 83)(17 41 106 204 155)(18 42 107 205 156)(19 43 108 206 157)(20 44 109 207 158)(21 45 110 208 159)(22 46 111 201 160)(23 47 112 202 153)(24 48 105 203 154)(25 149 173 36 165)(26 150 174 37 166)(27 151 175 38 167)(28 152 176 39 168)(29 145 169 40 161)(30 146 170 33 162)(31 147 171 34 163)(32 148 172 35 164)(49 185 222 238 135)(50 186 223 239 136)(51 187 224 240 129)(52 188 217 233 130)(53 189 218 234 131)(54 190 219 235 132)(55 191 220 236 133)(56 192 221 237 134)(89 113 214 99 198)(90 114 215 100 199)(91 115 216 101 200)(92 116 209 102 193)(93 117 210 103 194)(94 118 211 104 195)(95 119 212 97 196)(96 120 213 98 197)
(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 95)(2 94)(3 93)(4 92)(5 91)(6 90)(7 89)(8 96)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 112)(17 66)(18 65)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 132)(26 131)(27 130)(28 129)(29 136)(30 135)(31 134)(32 133)(33 222)(34 221)(35 220)(36 219)(37 218)(38 217)(39 224)(40 223)(41 81)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 146)(50 145)(51 152)(52 151)(53 150)(54 149)(55 148)(56 147)(57 97)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(73 213)(74 212)(75 211)(76 210)(77 209)(78 216)(79 215)(80 214)(113 179)(114 178)(115 177)(116 184)(117 183)(118 182)(119 181)(120 180)(121 200)(122 199)(123 198)(124 197)(125 196)(126 195)(127 194)(128 193)(137 203)(138 202)(139 201)(140 208)(141 207)(142 206)(143 205)(144 204)(153 229)(154 228)(155 227)(156 226)(157 225)(158 232)(159 231)(160 230)(161 239)(162 238)(163 237)(164 236)(165 235)(166 234)(167 233)(168 240)(169 186)(170 185)(171 192)(172 191)(173 190)(174 189)(175 188)(176 187)
(1 54 69)(2 55 70)(3 56 71)(4 49 72)(5 50 65)(6 51 66)(7 52 67)(8 53 68)(9 74 219)(10 75 220)(11 76 221)(12 77 222)(13 78 223)(14 79 224)(15 80 217)(16 73 218)(17 90 152)(18 91 145)(19 92 146)(20 93 147)(21 94 148)(22 95 149)(23 96 150)(24 89 151)(25 160 196)(26 153 197)(27 154 198)(28 155 199)(29 156 200)(30 157 193)(31 158 194)(32 159 195)(33 108 209)(34 109 210)(35 110 211)(36 111 212)(37 112 213)(38 105 214)(39 106 215)(40 107 216)(41 114 176)(42 115 169)(43 116 170)(44 117 171)(45 118 172)(46 119 173)(47 120 174)(48 113 175)(57 235 139)(58 236 140)(59 237 141)(60 238 142)(61 239 143)(62 240 144)(63 233 137)(64 234 138)(81 178 187)(82 179 188)(83 180 189)(84 181 190)(85 182 191)(86 183 192)(87 184 185)(88 177 186)(97 165 201)(98 166 202)(99 167 203)(100 168 204)(101 161 205)(102 162 206)(103 163 207)(104 164 208)(121 136 226)(122 129 227)(123 130 228)(124 131 229)(125 132 230)(126 133 231)(127 134 232)(128 135 225)
(9 219)(10 220)(11 221)(12 222)(13 223)(14 224)(15 217)(16 218)(17 148)(18 149)(19 150)(20 151)(21 152)(22 145)(23 146)(24 147)(25 156)(26 157)(27 158)(28 159)(29 160)(30 153)(31 154)(32 155)(33 112)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 172)(42 173)(43 174)(44 175)(45 176)(46 169)(47 170)(48 171)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(81 187)(82 188)(83 189)(84 190)(85 191)(86 192)(87 185)(88 186)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(113 117)(114 118)(115 119)(116 120)(129 227)(130 228)(131 229)(132 230)(133 231)(134 232)(135 225)(136 226)(137 233)(138 234)(139 235)(140 236)(141 237)(142 238)(143 239)(144 240)(161 201)(162 202)(163 203)(164 204)(165 205)(166 206)(167 207)(168 208)(193 197)(194 198)(195 199)(196 200)(209 213)(210 214)(211 215)(212 216)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10L 10M 10N 10O 10P 12 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20L 20M ··· 20T 24A 24B 30A 30B 30C 30D 30E ··· 30L 40A ··· 40H 40I ··· 40P 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 3 4 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 10 10 10 10 10 ··· 10 10 10 10 10 12 15 15 15 15 20 20 20 20 20 ··· 20 20 ··· 20 24 24 30 30 30 30 30 ··· 30 40 ··· 40 40 ··· 40 60 60 60 60 120 ··· 120 size 1 1 4 4 6 2 2 3 3 12 12 1 1 1 1 2 8 8 2 2 6 6 1 1 1 1 4 ··· 4 6 6 6 6 4 2 2 2 2 2 2 2 2 3 ··· 3 12 ··· 12 4 4 2 2 2 2 8 ··· 8 2 ··· 2 6 ··· 6 4 4 4 4 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 C5×S3 C4○D8 C5×D4 C5×D4 S3×C10 S3×C10 C5×C4○D8 S3×D4 D8⋊3S3 C5×S3×D4 C5×D8⋊3S3 kernel C5×D8⋊3S3 S3×C40 C5×Dic12 C5×D4.S3 C15×D8 C5×D4⋊2S3 D8⋊3S3 S3×C8 Dic12 D4.S3 C3×D8 D4⋊2S3 C5×D8 C5×Dic3 S3×C10 C40 C5×D4 D8 C15 Dic3 D6 C8 D4 C3 C10 C5 C2 C1 # reps 1 1 1 2 1 2 4 4 4 8 4 8 1 1 1 1 2 4 4 4 4 4 8 16 1 2 4 8

Matrix representation of C5×D83S3 in GL4(𝔽241) generated by

 98 0 0 0 0 98 0 0 0 0 87 0 0 0 0 87
,
 240 0 0 0 0 240 0 0 0 0 30 0 0 0 170 233
,
 1 0 0 0 0 1 0 0 0 0 82 217 0 0 9 159
,
 0 1 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 240 240 0 0 0 0 1 0 0 0 47 240
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,87,0,0,0,0,87],[240,0,0,0,0,240,0,0,0,0,30,170,0,0,0,233],[1,0,0,0,0,1,0,0,0,0,82,9,0,0,217,159],[0,240,0,0,1,240,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,47,0,0,0,240] >;

C5×D83S3 in GAP, Magma, Sage, TeX

C_5\times D_8\rtimes_3S_3
% in TeX

G:=Group("C5xD8:3S3");
// GroupNames label

G:=SmallGroup(480,791);
// by ID

G=gap.SmallGroup(480,791);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,1766,471,2111,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^3=e^2=1,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,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations

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