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## G = C5×S3×Q16order 480 = 25·3·5

### Direct product of C5, S3 and Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×S3×Q16
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×S3×Q8 — C5×S3×Q16
 Lower central C3 — C6 — C12 — C5×S3×Q16
 Upper central C1 — C10 — C20 — C5×Q16

Generators and relations for C5×S3×Q16
G = < a,b,c,d,e | a5=b3=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 260 in 120 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C5, S3 [×2], C6, C8, C8, C2×C4 [×3], Q8 [×2], Q8 [×4], C10, C10 [×2], Dic3, Dic3 [×2], C12, C12 [×2], D6, C15, C2×C8, Q16, Q16 [×3], C2×Q8 [×2], C20, C20 [×5], C2×C10, C3⋊C8, C24, Dic6 [×2], Dic6 [×2], C4×S3, C4×S3 [×2], C3×Q8 [×2], C5×S3 [×2], C30, C2×Q16, C40, C40, C2×C20 [×3], C5×Q8 [×2], C5×Q8 [×4], S3×C8, Dic12, C3⋊Q16 [×2], C3×Q16, S3×Q8 [×2], C5×Dic3, C5×Dic3 [×2], C60, C60 [×2], S3×C10, C2×C40, C5×Q16, C5×Q16 [×3], Q8×C10 [×2], S3×Q16, C5×C3⋊C8, C120, C5×Dic6 [×2], C5×Dic6 [×2], S3×C20, S3×C20 [×2], Q8×C15 [×2], C10×Q16, S3×C40, C5×Dic12, C5×C3⋊Q16 [×2], C15×Q16, C5×S3×Q8 [×2], C5×S3×Q16
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], Q16 [×2], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C2×Q16, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], C5×Q16 [×2], D4×C10, S3×Q16, S3×C2×C10, C10×Q16, C5×S3×D4, C5×S3×Q16

Smallest permutation representation of C5×S3×Q16
On 240 points
Generators in S240
(1 181 79 57 127)(2 182 80 58 128)(3 183 73 59 121)(4 184 74 60 122)(5 177 75 61 123)(6 178 76 62 124)(7 179 77 63 125)(8 180 78 64 126)(9 133 230 65 84)(10 134 231 66 85)(11 135 232 67 86)(12 136 225 68 87)(13 129 226 69 88)(14 130 227 70 81)(15 131 228 71 82)(16 132 229 72 83)(17 41 110 204 159)(18 42 111 205 160)(19 43 112 206 153)(20 44 105 207 154)(21 45 106 208 155)(22 46 107 201 156)(23 47 108 202 157)(24 48 109 203 158)(25 145 169 36 161)(26 146 170 37 162)(27 147 171 38 163)(28 148 172 39 164)(29 149 173 40 165)(30 150 174 33 166)(31 151 175 34 167)(32 152 176 35 168)(49 185 222 238 144)(50 186 223 239 137)(51 187 224 240 138)(52 188 217 233 139)(53 189 218 234 140)(54 190 219 235 141)(55 191 220 236 142)(56 192 221 237 143)(89 113 215 101 199)(90 114 216 102 200)(91 115 209 103 193)(92 116 210 104 194)(93 117 211 97 195)(94 118 212 98 196)(95 119 213 99 197)(96 120 214 100 198)
(1 65 54)(2 66 55)(3 67 56)(4 68 49)(5 69 50)(6 70 51)(7 71 52)(8 72 53)(9 219 79)(10 220 80)(11 221 73)(12 222 74)(13 223 75)(14 224 76)(15 217 77)(16 218 78)(17 148 89)(18 149 90)(19 150 91)(20 151 92)(21 152 93)(22 145 94)(23 146 95)(24 147 96)(25 196 156)(26 197 157)(27 198 158)(28 199 159)(29 200 160)(30 193 153)(31 194 154)(32 195 155)(33 209 112)(34 210 105)(35 211 106)(36 212 107)(37 213 108)(38 214 109)(39 215 110)(40 216 111)(41 172 113)(42 173 114)(43 174 115)(44 175 116)(45 176 117)(46 169 118)(47 170 119)(48 171 120)(57 133 235)(58 134 236)(59 135 237)(60 136 238)(61 129 239)(62 130 240)(63 131 233)(64 132 234)(81 187 178)(82 188 179)(83 189 180)(84 190 181)(85 191 182)(86 192 183)(87 185 184)(88 186 177)(97 208 168)(98 201 161)(99 202 162)(100 203 163)(101 204 164)(102 205 165)(103 206 166)(104 207 167)(121 232 143)(122 225 144)(123 226 137)(124 227 138)(125 228 139)(126 229 140)(127 230 141)(128 231 142)
(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 68)(50 69)(51 70)(52 71)(53 72)(54 65)(55 66)(56 67)(81 187)(82 188)(83 189)(84 190)(85 191)(86 192)(87 185)(88 186)(129 239)(130 240)(131 233)(132 234)(133 235)(134 236)(135 237)(136 238)(137 226)(138 227)(139 228)(140 229)(141 230)(142 231)(143 232)(144 225)(161 201)(162 202)(163 203)(164 204)(165 205)(166 206)(167 207)(168 208)
(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 5 91)(2 94 6 90)(3 93 7 89)(4 92 8 96)(9 108 13 112)(10 107 14 111)(11 106 15 110)(12 105 16 109)(17 67 21 71)(18 66 22 70)(19 65 23 69)(20 72 24 68)(25 138 29 142)(26 137 30 141)(27 144 31 140)(28 143 32 139)(33 219 37 223)(34 218 38 222)(35 217 39 221)(36 224 40 220)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)(49 151 53 147)(50 150 54 146)(51 149 55 145)(52 148 56 152)(57 99 61 103)(58 98 62 102)(59 97 63 101)(60 104 64 100)(73 211 77 215)(74 210 78 214)(75 209 79 213)(76 216 80 212)(113 183 117 179)(114 182 118 178)(115 181 119 177)(116 180 120 184)(121 195 125 199)(122 194 126 198)(123 193 127 197)(124 200 128 196)(129 206 133 202)(130 205 134 201)(131 204 135 208)(132 203 136 207)(153 230 157 226)(154 229 158 225)(155 228 159 232)(156 227 160 231)(161 240 165 236)(162 239 166 235)(163 238 167 234)(164 237 168 233)(169 187 173 191)(170 186 174 190)(171 185 175 189)(172 192 176 188)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10L 12A 12B 12C 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20L 20M 20N 20O 20P 20Q ··· 20X 24A 24B 30A 30B 30C 30D 40A ··· 40H 40I ··· 40P 60A 60B 60C 60D 60E ··· 60L 120A ··· 120H order 1 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 8 8 8 8 10 10 10 10 10 ··· 10 12 12 12 15 15 15 15 20 20 20 20 20 ··· 20 20 20 20 20 20 ··· 20 24 24 30 30 30 30 40 ··· 40 40 ··· 40 60 60 60 60 60 ··· 60 120 ··· 120 size 1 1 3 3 2 2 4 4 6 12 12 1 1 1 1 2 2 2 6 6 1 1 1 1 3 ··· 3 4 8 8 2 2 2 2 2 2 2 2 4 ··· 4 6 6 6 6 12 ··· 12 4 4 2 2 2 2 2 ··· 2 6 ··· 6 4 4 4 4 8 ··· 8 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 Q16 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×Q16 S3×D4 S3×Q16 C5×S3×D4 C5×S3×Q16 kernel C5×S3×Q16 S3×C40 C5×Dic12 C5×C3⋊Q16 C15×Q16 C5×S3×Q8 S3×Q16 S3×C8 Dic12 C3⋊Q16 C3×Q16 S3×Q8 C5×Q16 C5×Dic3 S3×C10 C40 C5×Q8 C5×S3 Q16 Dic3 D6 C8 Q8 S3 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×S3×Q16 in GL4(𝔽241) generated by

 98 0 0 0 0 98 0 0 0 0 1 0 0 0 0 1
,
 240 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 73 0 0 33 219
,
 240 0 0 0 0 240 0 0 0 0 165 195 0 0 47 76
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,1,0,0,0,0,1],[240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,33,0,0,73,219],[240,0,0,0,0,240,0,0,0,0,165,47,0,0,195,76] >;

C5×S3×Q16 in GAP, Magma, Sage, TeX

C_5\times S_3\times Q_{16}
% in TeX

G:=Group("C5xS3xQ16");
// GroupNames label

G:=SmallGroup(480,796);
// by ID

G=gap.SmallGroup(480,796);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,471,436,2111,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^8=1,e^2=d^4,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