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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 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3 [×2], C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×2], C10, C10 [×2], Dic3, Dic3, C12, C12 [×2], D6, D6, C15, M4(2), SD16 [×2], Q16, Q16, C2×Q8, C4○D4, C20, C20 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3 [×2], D12, D12, C3×Q8 [×2], C5×S3 [×2], C30, C8.C22, C40, C40, C2×C20 [×3], C5×D4 [×2], C5×Q8 [×2], C5×Q8 [×2], C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, S3×Q8, Q83S3, C5×Dic3, C5×Dic3, C60, C60 [×2], S3×C10, S3×C10, C5×M4(2), C5×SD16 [×2], C5×Q16, C5×Q16, Q8×C10, C5×C4○D4, Q16⋊S3, C5×C3⋊C8, C120, C5×Dic6, C5×Dic6, S3×C20, S3×C20 [×2], C5×D12, C5×D12, Q8×C15 [×2], 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 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C8.C22, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], 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
(1 181 79 57 121)(2 182 80 58 122)(3 183 73 59 123)(4 184 74 60 124)(5 177 75 61 125)(6 178 76 62 126)(7 179 77 63 127)(8 180 78 64 128)(9 133 50 68 84)(10 134 51 69 85)(11 135 52 70 86)(12 136 53 71 87)(13 129 54 72 88)(14 130 55 65 81)(15 131 56 66 82)(16 132 49 67 83)(17 41 110 32 159)(18 42 111 25 160)(19 43 112 26 153)(20 44 105 27 154)(21 45 106 28 155)(22 46 107 29 156)(23 47 108 30 157)(24 48 109 31 158)(33 166 202 150 174)(34 167 203 151 175)(35 168 204 152 176)(36 161 205 145 169)(37 162 206 146 170)(38 163 207 147 171)(39 164 208 148 172)(40 165 201 149 173)(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)(137 234 218 225 189)(138 235 219 226 190)(139 236 220 227 191)(140 237 221 228 192)(141 238 222 229 185)(142 239 223 230 186)(143 240 224 231 187)(144 233 217 232 188)
(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 70 21 66)(18 69 22 65)(19 68 23 72)(20 67 24 71)(25 134 29 130)(26 133 30 129)(27 132 31 136)(28 131 32 135)(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)(49 158 53 154)(50 157 54 153)(51 156 55 160)(52 155 56 159)(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)(121 197 125 193)(122 196 126 200)(123 195 127 199)(124 194 128 198)(145 231 149 227)(146 230 150 226)(147 229 151 225)(148 228 152 232)(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)(201 220 205 224)(202 219 206 223)(203 218 207 222)(204 217 208 221)
(1 226 68)(2 227 69)(3 228 70)(4 229 71)(5 230 72)(6 231 65)(7 232 66)(8 225 67)(9 79 138)(10 80 139)(11 73 140)(12 74 141)(13 75 142)(14 76 143)(15 77 144)(16 78 137)(17 89 148)(18 90 149)(19 91 150)(20 92 151)(21 93 152)(22 94 145)(23 95 146)(24 96 147)(25 102 165)(26 103 166)(27 104 167)(28 97 168)(29 98 161)(30 99 162)(31 100 163)(32 101 164)(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)(49 128 218)(50 121 219)(51 122 220)(52 123 221)(53 124 222)(54 125 223)(55 126 224)(56 127 217)(57 235 133)(58 236 134)(59 237 135)(60 238 136)(61 239 129)(62 240 130)(63 233 131)(64 234 132)(81 178 187)(82 179 188)(83 180 189)(84 181 190)(85 182 191)(86 183 192)(87 184 185)(88 177 186)(153 193 202)(154 194 203)(155 195 204)(156 196 205)(157 197 206)(158 198 207)(159 199 208)(160 200 201)
(2 6)(4 8)(9 138)(10 143)(11 140)(12 137)(13 142)(14 139)(15 144)(16 141)(17 152)(18 149)(19 146)(20 151)(21 148)(22 145)(23 150)(24 147)(25 165)(26 162)(27 167)(28 164)(29 161)(30 166)(31 163)(32 168)(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)(49 222)(50 219)(51 224)(52 221)(53 218)(54 223)(55 220)(56 217)(58 62)(60 64)(65 227)(66 232)(67 229)(68 226)(69 231)(70 228)(71 225)(72 230)(74 78)(76 80)(81 191)(82 188)(83 185)(84 190)(85 187)(86 192)(87 189)(88 186)(89 93)(91 95)(97 101)(99 103)(113 117)(115 119)(122 126)(124 128)(129 239)(130 236)(131 233)(132 238)(133 235)(134 240)(135 237)(136 234)(153 206)(154 203)(155 208)(156 205)(157 202)(158 207)(159 204)(160 201)(178 182)(180 184)(193 197)(195 199)(209 213)(211 215)

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

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

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

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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