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## G = C5×Q16⋊S3order 480 = 25·3·5

### Direct product of C5 and Q16⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×Q16⋊S3
G = < a,b,c,d,e | a5=b8=d3=e2=1, c2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 292 in 120 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C8, C2×C4, D4, Q8, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C15, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C3×Q8, C5×S3, C30, C8.C22, C40, C40, C2×C20, C5×D4, C5×Q8, C5×Q8, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, S3×Q8, Q83S3, C5×Dic3, C5×Dic3, C60, C60, S3×C10, S3×C10, C5×M4(2), C5×SD16, C5×Q16, C5×Q16, Q8×C10, C5×C4○D4, Q16⋊S3, C5×C3⋊C8, C120, C5×Dic6, C5×Dic6, S3×C20, S3×C20, C5×D12, C5×D12, Q8×C15, C5×C8.C22, C5×C8⋊S3, C5×C24⋊C2, C5×Q82S3, C5×C3⋊Q16, C15×Q16, C5×S3×Q8, C5×Q83S3, C5×Q16⋊S3
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C8.C22, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, Q16⋊S3, S3×C2×C10, C5×C8.C22, C5×S3×D4, C5×Q16⋊S3

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

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C8.C22 S3×D4 Q16⋊S3 C5×C8.C22 C5×S3×D4 C5×Q16⋊S3 kernel C5×Q16⋊S3 C5×C8⋊S3 C5×C24⋊C2 C5×Q8⋊2S3 C5×C3⋊Q16 C15×Q16 C5×S3×Q8 C5×Q8⋊3S3 Q16⋊S3 C8⋊S3 C24⋊C2 Q8⋊2S3 C3⋊Q16 C3×Q16 S3×Q8 Q8⋊3S3 C5×Q16 C5×Dic3 S3×C10 C40 C5×Q8 Q16 Dic3 D6 C8 Q8 C15 C10 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 1 1 1 1 2 4 4 4 4 8 1 1 2 4 4 8

Matrix representation of C5×Q16⋊S3 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 87 0 0 0 0 0 0 87 0 0 0 0 0 0 87 0 0 0 0 0 0 87
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 77 143 0 0 199 0 190 236 0 0 40 109 70 130 0 0 179 93 37 172
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 227 0 160 0 0 0 228 0 0 1 0 0 175 0 14 0 0 0 59 240 152 0
,
 240 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 199 0 240 0 0 0 26 0 0 240

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,87,0,0,0,0,0,0,87,0,0,0,0,0,0,87,0,0,0,0,0,0,87],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,199,40,179,0,0,0,0,109,93,0,0,77,190,70,37,0,0,143,236,130,172],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,227,228,175,59,0,0,0,0,0,240,0,0,160,0,14,152,0,0,0,1,0,0],[240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,199,26,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;

C5×Q16⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C5xQ16:S3");
// GroupNames label

G:=SmallGroup(480,797);
// by ID

G=gap.SmallGroup(480,797);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,1766,471,436,2111,1068,102,15686]);
// Polycyclic

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

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