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G = S3×C5⋊Q16order 480 = 25·3·5

Direct product of S3 and C5⋊Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×C5⋊Q16, C60.33C23, Dic10.25D6, Dic6.11D10, Dic30.10C22, C55(S3×Q16), (S3×Q8).D5, C158(C2×Q16), (C5×S3)⋊2Q16, Q8.8(S3×D5), C15⋊Q166C2, C5⋊Dic125C2, C157Q163C2, C52C8.17D6, (C3×Q8).1D10, (C5×Q8).35D6, (C4×S3).24D10, (S3×C10).36D4, C10.149(S3×D4), C30.195(C2×D4), D6.22(C5⋊D4), C20.33(C22×S3), C153C8.9C22, (C5×Dic3).16D4, (S3×Dic10).1C2, C12.33(C22×D5), (Q8×C15).3C22, (S3×C20).11C22, Dic3.6(C5⋊D4), (C5×Dic6).10C22, (C3×Dic10).9C22, C4.33(C2×S3×D5), C32(C2×C5⋊Q16), (C5×S3×Q8).1C2, (S3×C52C8).1C2, (C3×C5⋊Q16)⋊1C2, C2.30(S3×C5⋊D4), C6.52(C2×C5⋊D4), (C3×C52C8).7C22, SmallGroup(480,585)

Series: Derived Chief Lower central Upper central

C1C60 — S3×C5⋊Q16
C1C5C15C30C60C3×Dic10S3×Dic10 — S3×C5⋊Q16
C15C30C60 — S3×C5⋊Q16
C1C2C4Q8

Generators and relations for S3×C5⋊Q16
 G = < a,b,c,d,e | a3=b2=c5=d8=1, e2=d4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 524 in 120 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C5, S3 [×2], C6, C8 [×2], C2×C4 [×3], Q8, Q8 [×5], C10, C10 [×2], Dic3, Dic3 [×2], C12, C12 [×2], D6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×2], C20, C20 [×3], C2×C10, C3⋊C8, C24, Dic6, Dic6 [×3], C4×S3, C4×S3 [×2], C3×Q8, C3×Q8, C5×S3 [×2], C30, C2×Q16, C52C8, C52C8, Dic10, Dic10 [×2], C2×Dic5, C2×C20 [×2], C5×Q8, C5×Q8 [×2], S3×C8, Dic12, C3⋊Q16 [×2], C3×Q16, S3×Q8, S3×Q8, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C60, S3×C10, C2×C52C8, C5⋊Q16, C5⋊Q16 [×3], C2×Dic10, Q8×C10, S3×Q16, C3×C52C8, C153C8, S3×Dic5, C15⋊Q8, C3×Dic10, C5×Dic6, C5×Dic6, S3×C20, S3×C20, Dic30, Q8×C15, C2×C5⋊Q16, S3×C52C8, C15⋊Q16, C5⋊Dic12, C3×C5⋊Q16, C157Q16, S3×Dic10, C5×S3×Q8, S3×C5⋊Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], C22×S3, C2×Q16, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C5⋊Q16 [×2], C2×C5⋊D4, S3×Q16, C2×S3×D5, C2×C5⋊Q16, S3×C5⋊D4, S3×C5⋊Q16

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

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

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

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

51 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B 6 8A8B8C8D10A10B10C10D10E10F12A12B12C15A15B20A···20F20G···20L24A24B30A30B60A···60F
order122234444445568888101010101010121212151520···2020···202424303060···60
size11332246122060222101030302266664840444···412···122020448···8

51 irreducible representations

dim1111111122222222222224444448
type+++++++++++++++-+++++--+-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6Q16D10D10D10C5⋊D4C5⋊D4S3×D4S3×D5C5⋊Q16S3×Q16C2×S3×D5S3×C5⋊D4S3×C5⋊Q16
kernelS3×C5⋊Q16S3×C52C8C15⋊Q16C5⋊Dic12C3×C5⋊Q16C157Q16S3×Dic10C5×S3×Q8C5⋊Q16C5×Dic3S3×C10S3×Q8C52C8Dic10C5×Q8C5×S3Dic6C4×S3C3×Q8Dic3D6C10Q8S3C5C4C2C1
# reps1111111111121114222441242242

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

100000
010000
00024000
00124000
000010
000001
,
100000
010000
000100
001000
00002400
00000240
,
9100000
239980000
001000
000100
000010
000001
,
60310000
1561810000
00240000
00024000
00000219
000011219
,
24000000
02400000
00240000
00024000
00001118
000020230

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[91,239,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,156,0,0,0,0,31,181,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,11,0,0,0,0,219,219],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,11,20,0,0,0,0,18,230] >;

S3×C5⋊Q16 in GAP, Magma, Sage, TeX

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

G:=Group("S3xC5:Q16");
// GroupNames label

G:=SmallGroup(480,585);
// by ID

G=gap.SmallGroup(480,585);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,100,675,185,80,1356,18822]);
// Polycyclic

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

׿
×
𝔽