direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×S3×Q16, C40.61D6, Dic12⋊5C10, C60.224C23, C120.68C22, C3⋊2(C10×Q16), (S3×Q8).C10, C8.9(S3×C10), C15⋊15(C2×Q16), (S3×C8).1C10, (S3×C40).4C2, C24.7(C2×C10), (C15×Q16)⋊9C2, (C3×Q16)⋊2C10, C3⋊Q16⋊3C10, D6.14(C5×D4), C6.34(D4×C10), Q8.8(S3×C10), (C5×Q8).46D6, (S3×C10).50D4, C30.370(C2×D4), C10.188(S3×D4), Dic3.5(C5×D4), (C5×Dic12)⋊13C2, C12.8(C22×C10), Dic6.4(C2×C10), (C5×Dic3).32D4, (S3×C20).60C22, C20.197(C22×S3), (Q8×C15).34C22, (C5×Dic6).35C22, C4.8(S3×C2×C10), C2.22(C5×S3×D4), (C5×S3×Q8).2C2, C3⋊C8.7(C2×C10), (C5×C3⋊Q16)⋊11C2, (C5×C3⋊C8).43C22, (C3×Q8).3(C2×C10), (C4×S3).11(C2×C10), SmallGroup(480,796)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C5, S3, C6, C8, C8, C2×C4, Q8, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, C15, C2×C8, Q16, Q16, C2×Q8, C20, C20, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C3×Q8, C5×S3, C30, C2×Q16, C40, C40, C2×C20, C5×Q8, C5×Q8, S3×C8, Dic12, C3⋊Q16, C3×Q16, S3×Q8, C5×Dic3, C5×Dic3, C60, C60, S3×C10, C2×C40, C5×Q16, C5×Q16, Q8×C10, S3×Q16, C5×C3⋊C8, C120, C5×Dic6, C5×Dic6, S3×C20, S3×C20, Q8×C15, C10×Q16, S3×C40, C5×Dic12, C5×C3⋊Q16, C15×Q16, C5×S3×Q8, C5×S3×Q16
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, Q16, C2×D4, C2×C10, C22×S3, C5×S3, C2×Q16, C5×D4, C22×C10, S3×D4, S3×C10, C5×Q16, D4×C10, S3×Q16, S3×C2×C10, C10×Q16, C5×S3×D4, C5×S3×Q16
►On 240 points
Generators in S240
(1 178 221 57 75)(2 179 222 58 76)(3 180 223 59 77)(4 181 224 60 78)(5 182 217 61 79)(6 183 218 62 80)(7 184 219 63 73)(8 177 220 64 74)(9 133 53 68 84)(10 134 54 69 85)(11 135 55 70 86)(12 136 56 71 87)(13 129 49 72 88)(14 130 50 65 81)(15 131 51 66 82)(16 132 52 67 83)(17 41 110 27 159)(18 42 111 28 160)(19 43 112 29 153)(20 44 105 30 154)(21 45 106 31 155)(22 46 107 32 156)(23 47 108 25 157)(24 48 109 26 158)(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)(89 113 215 104 199)(90 114 216 97 200)(91 115 209 98 193)(92 116 210 99 194)(93 117 211 100 195)(94 118 212 101 196)(95 119 213 102 197)(96 120 214 103 198)(121 225 189 137 234)(122 226 190 138 235)(123 227 191 139 236)(124 228 192 140 237)(125 229 185 141 238)(126 230 186 142 239)(127 231 187 143 240)(128 232 188 144 233)
(1 68 226)(2 69 227)(3 70 228)(4 71 229)(5 72 230)(6 65 231)(7 66 232)(8 67 225)(9 138 221)(10 139 222)(11 140 223)(12 141 224)(13 142 217)(14 143 218)(15 144 219)(16 137 220)(17 148 89)(18 149 90)(19 150 91)(20 151 92)(21 152 93)(22 145 94)(23 146 95)(24 147 96)(25 162 102)(26 163 103)(27 164 104)(28 165 97)(29 166 98)(30 167 99)(31 168 100)(32 161 101)(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)(49 126 79)(50 127 80)(51 128 73)(52 121 74)(53 122 75)(54 123 76)(55 124 77)(56 125 78)(57 133 235)(58 134 236)(59 135 237)(60 136 238)(61 129 239)(62 130 240)(63 131 233)(64 132 234)(81 187 183)(82 188 184)(83 189 177)(84 190 178)(85 191 179)(86 192 180)(87 185 181)(88 186 182)(153 202 193)(154 203 194)(155 204 195)(156 205 196)(157 206 197)(158 207 198)(159 208 199)(160 201 200)
(9 138)(10 139)(11 140)(12 141)(13 142)(14 143)(15 144)(16 137)(17 148)(18 149)(19 150)(20 151)(21 152)(22 145)(23 146)(24 147)(25 162)(26 163)(27 164)(28 165)(29 166)(30 167)(31 168)(32 161)(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)(49 126)(50 127)(51 128)(52 121)(53 122)(54 123)(55 124)(56 125)(65 231)(66 232)(67 225)(68 226)(69 227)(70 228)(71 229)(72 230)(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)(153 202)(154 203)(155 204)(156 205)(157 206)(158 207)(159 208)(160 201)
(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)(25 129 29 133)(26 136 30 132)(27 135 31 131)(28 134 32 130)(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)(49 153 53 157)(50 160 54 156)(51 159 55 155)(52 158 56 154)(57 102 61 98)(58 101 62 97)(59 100 63 104)(60 99 64 103)(73 199 77 195)(74 198 78 194)(75 197 79 193)(76 196 80 200)(113 180 117 184)(114 179 118 183)(115 178 119 182)(116 177 120 181)(121 207 125 203)(122 206 126 202)(123 205 127 201)(124 204 128 208)(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)(209 221 213 217)(210 220 214 224)(211 219 215 223)(212 218 216 222)
G:=sub<Sym(240)| (1,178,221,57,75)(2,179,222,58,76)(3,180,223,59,77)(4,181,224,60,78)(5,182,217,61,79)(6,183,218,62,80)(7,184,219,63,73)(8,177,220,64,74)(9,133,53,68,84)(10,134,54,69,85)(11,135,55,70,86)(12,136,56,71,87)(13,129,49,72,88)(14,130,50,65,81)(15,131,51,66,82)(16,132,52,67,83)(17,41,110,27,159)(18,42,111,28,160)(19,43,112,29,153)(20,44,105,30,154)(21,45,106,31,155)(22,46,107,32,156)(23,47,108,25,157)(24,48,109,26,158)(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)(89,113,215,104,199)(90,114,216,97,200)(91,115,209,98,193)(92,116,210,99,194)(93,117,211,100,195)(94,118,212,101,196)(95,119,213,102,197)(96,120,214,103,198)(121,225,189,137,234)(122,226,190,138,235)(123,227,191,139,236)(124,228,192,140,237)(125,229,185,141,238)(126,230,186,142,239)(127,231,187,143,240)(128,232,188,144,233), (1,68,226)(2,69,227)(3,70,228)(4,71,229)(5,72,230)(6,65,231)(7,66,232)(8,67,225)(9,138,221)(10,139,222)(11,140,223)(12,141,224)(13,142,217)(14,143,218)(15,144,219)(16,137,220)(17,148,89)(18,149,90)(19,150,91)(20,151,92)(21,152,93)(22,145,94)(23,146,95)(24,147,96)(25,162,102)(26,163,103)(27,164,104)(28,165,97)(29,166,98)(30,167,99)(31,168,100)(32,161,101)(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)(49,126,79)(50,127,80)(51,128,73)(52,121,74)(53,122,75)(54,123,76)(55,124,77)(56,125,78)(57,133,235)(58,134,236)(59,135,237)(60,136,238)(61,129,239)(62,130,240)(63,131,233)(64,132,234)(81,187,183)(82,188,184)(83,189,177)(84,190,178)(85,191,179)(86,192,180)(87,185,181)(88,186,182)(153,202,193)(154,203,194)(155,204,195)(156,205,196)(157,206,197)(158,207,198)(159,208,199)(160,201,200), (9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,137)(17,148)(18,149)(19,150)(20,151)(21,152)(22,145)(23,146)(24,147)(25,162)(26,163)(27,164)(28,165)(29,166)(30,167)(31,168)(32,161)(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)(49,126)(50,127)(51,128)(52,121)(53,122)(54,123)(55,124)(56,125)(65,231)(66,232)(67,225)(68,226)(69,227)(70,228)(71,229)(72,230)(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)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,201), (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)(25,129,29,133)(26,136,30,132)(27,135,31,131)(28,134,32,130)(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)(49,153,53,157)(50,160,54,156)(51,159,55,155)(52,158,56,154)(57,102,61,98)(58,101,62,97)(59,100,63,104)(60,99,64,103)(73,199,77,195)(74,198,78,194)(75,197,79,193)(76,196,80,200)(113,180,117,184)(114,179,118,183)(115,178,119,182)(116,177,120,181)(121,207,125,203)(122,206,126,202)(123,205,127,201)(124,204,128,208)(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)(209,221,213,217)(210,220,214,224)(211,219,215,223)(212,218,216,222)>;
G:=Group( (1,178,221,57,75)(2,179,222,58,76)(3,180,223,59,77)(4,181,224,60,78)(5,182,217,61,79)(6,183,218,62,80)(7,184,219,63,73)(8,177,220,64,74)(9,133,53,68,84)(10,134,54,69,85)(11,135,55,70,86)(12,136,56,71,87)(13,129,49,72,88)(14,130,50,65,81)(15,131,51,66,82)(16,132,52,67,83)(17,41,110,27,159)(18,42,111,28,160)(19,43,112,29,153)(20,44,105,30,154)(21,45,106,31,155)(22,46,107,32,156)(23,47,108,25,157)(24,48,109,26,158)(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)(89,113,215,104,199)(90,114,216,97,200)(91,115,209,98,193)(92,116,210,99,194)(93,117,211,100,195)(94,118,212,101,196)(95,119,213,102,197)(96,120,214,103,198)(121,225,189,137,234)(122,226,190,138,235)(123,227,191,139,236)(124,228,192,140,237)(125,229,185,141,238)(126,230,186,142,239)(127,231,187,143,240)(128,232,188,144,233), (1,68,226)(2,69,227)(3,70,228)(4,71,229)(5,72,230)(6,65,231)(7,66,232)(8,67,225)(9,138,221)(10,139,222)(11,140,223)(12,141,224)(13,142,217)(14,143,218)(15,144,219)(16,137,220)(17,148,89)(18,149,90)(19,150,91)(20,151,92)(21,152,93)(22,145,94)(23,146,95)(24,147,96)(25,162,102)(26,163,103)(27,164,104)(28,165,97)(29,166,98)(30,167,99)(31,168,100)(32,161,101)(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)(49,126,79)(50,127,80)(51,128,73)(52,121,74)(53,122,75)(54,123,76)(55,124,77)(56,125,78)(57,133,235)(58,134,236)(59,135,237)(60,136,238)(61,129,239)(62,130,240)(63,131,233)(64,132,234)(81,187,183)(82,188,184)(83,189,177)(84,190,178)(85,191,179)(86,192,180)(87,185,181)(88,186,182)(153,202,193)(154,203,194)(155,204,195)(156,205,196)(157,206,197)(158,207,198)(159,208,199)(160,201,200), (9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,137)(17,148)(18,149)(19,150)(20,151)(21,152)(22,145)(23,146)(24,147)(25,162)(26,163)(27,164)(28,165)(29,166)(30,167)(31,168)(32,161)(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)(49,126)(50,127)(51,128)(52,121)(53,122)(54,123)(55,124)(56,125)(65,231)(66,232)(67,225)(68,226)(69,227)(70,228)(71,229)(72,230)(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)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,201), (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)(25,129,29,133)(26,136,30,132)(27,135,31,131)(28,134,32,130)(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)(49,153,53,157)(50,160,54,156)(51,159,55,155)(52,158,56,154)(57,102,61,98)(58,101,62,97)(59,100,63,104)(60,99,64,103)(73,199,77,195)(74,198,78,194)(75,197,79,193)(76,196,80,200)(113,180,117,184)(114,179,118,183)(115,178,119,182)(116,177,120,181)(121,207,125,203)(122,206,126,202)(123,205,127,201)(124,204,128,208)(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)(209,221,213,217)(210,220,214,224)(211,219,215,223)(212,218,216,222) );
G=PermutationGroup([[(1,178,221,57,75),(2,179,222,58,76),(3,180,223,59,77),(4,181,224,60,78),(5,182,217,61,79),(6,183,218,62,80),(7,184,219,63,73),(8,177,220,64,74),(9,133,53,68,84),(10,134,54,69,85),(11,135,55,70,86),(12,136,56,71,87),(13,129,49,72,88),(14,130,50,65,81),(15,131,51,66,82),(16,132,52,67,83),(17,41,110,27,159),(18,42,111,28,160),(19,43,112,29,153),(20,44,105,30,154),(21,45,106,31,155),(22,46,107,32,156),(23,47,108,25,157),(24,48,109,26,158),(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),(89,113,215,104,199),(90,114,216,97,200),(91,115,209,98,193),(92,116,210,99,194),(93,117,211,100,195),(94,118,212,101,196),(95,119,213,102,197),(96,120,214,103,198),(121,225,189,137,234),(122,226,190,138,235),(123,227,191,139,236),(124,228,192,140,237),(125,229,185,141,238),(126,230,186,142,239),(127,231,187,143,240),(128,232,188,144,233)], [(1,68,226),(2,69,227),(3,70,228),(4,71,229),(5,72,230),(6,65,231),(7,66,232),(8,67,225),(9,138,221),(10,139,222),(11,140,223),(12,141,224),(13,142,217),(14,143,218),(15,144,219),(16,137,220),(17,148,89),(18,149,90),(19,150,91),(20,151,92),(21,152,93),(22,145,94),(23,146,95),(24,147,96),(25,162,102),(26,163,103),(27,164,104),(28,165,97),(29,166,98),(30,167,99),(31,168,100),(32,161,101),(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),(49,126,79),(50,127,80),(51,128,73),(52,121,74),(53,122,75),(54,123,76),(55,124,77),(56,125,78),(57,133,235),(58,134,236),(59,135,237),(60,136,238),(61,129,239),(62,130,240),(63,131,233),(64,132,234),(81,187,183),(82,188,184),(83,189,177),(84,190,178),(85,191,179),(86,192,180),(87,185,181),(88,186,182),(153,202,193),(154,203,194),(155,204,195),(156,205,196),(157,206,197),(158,207,198),(159,208,199),(160,201,200)], [(9,138),(10,139),(11,140),(12,141),(13,142),(14,143),(15,144),(16,137),(17,148),(18,149),(19,150),(20,151),(21,152),(22,145),(23,146),(24,147),(25,162),(26,163),(27,164),(28,165),(29,166),(30,167),(31,168),(32,161),(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),(49,126),(50,127),(51,128),(52,121),(53,122),(54,123),(55,124),(56,125),(65,231),(66,232),(67,225),(68,226),(69,227),(70,228),(71,229),(72,230),(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),(153,202),(154,203),(155,204),(156,205),(157,206),(158,207),(159,208),(160,201)], [(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),(25,129,29,133),(26,136,30,132),(27,135,31,131),(28,134,32,130),(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),(49,153,53,157),(50,160,54,156),(51,159,55,155),(52,158,56,154),(57,102,61,98),(58,101,62,97),(59,100,63,104),(60,99,64,103),(73,199,77,195),(74,198,78,194),(75,197,79,193),(76,196,80,200),(113,180,117,184),(114,179,118,183),(115,178,119,182),(116,177,120,181),(121,207,125,203),(122,206,126,202),(123,205,127,201),(124,204,128,208),(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),(209,221,213,217),(210,220,214,224),(211,219,215,223),(212,218,216,222)]])
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