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## G = S3×C5⋊Q16order 480 = 25·3·5

### Direct product of S3 and C5⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — S3×C5⋊Q16
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — S3×Dic10 — S3×C5⋊Q16
 Lower central C15 — C30 — C60 — S3×C5⋊Q16
 Upper central C1 — C2 — C4 — Q8

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, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, Q8, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, C15, C2×C8, Q16, C2×Q8, Dic5, C20, C20, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C3×Q8, C3×Q8, C5×S3, C30, C2×Q16, C52C8, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C5×Q8, C5×Q8, S3×C8, Dic12, C3⋊Q16, C3×Q16, S3×Q8, S3×Q8, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C60, S3×C10, C2×C52C8, C5⋊Q16, C5⋊Q16, 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, C22, S3, D4, C23, D5, D6, Q16, C2×D4, D10, C22×S3, C2×Q16, C5⋊D4, C22×D5, S3×D4, S3×D5, C5⋊Q16, 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 45 95)(2 46 96)(3 47 89)(4 48 90)(5 41 91)(6 42 92)(7 43 93)(8 44 94)(9 121 184)(10 122 177)(11 123 178)(12 124 179)(13 125 180)(14 126 181)(15 127 182)(16 128 183)(17 170 71)(18 171 72)(19 172 65)(20 173 66)(21 174 67)(22 175 68)(23 176 69)(24 169 70)(25 236 107)(26 237 108)(27 238 109)(28 239 110)(29 240 111)(30 233 112)(31 234 105)(32 235 106)(33 151 58)(34 152 59)(35 145 60)(36 146 61)(37 147 62)(38 148 63)(39 149 64)(40 150 57)(49 224 129)(50 217 130)(51 218 131)(52 219 132)(53 220 133)(54 221 134)(55 222 135)(56 223 136)(73 157 194)(74 158 195)(75 159 196)(76 160 197)(77 153 198)(78 154 199)(79 155 200)(80 156 193)(81 162 204)(82 163 205)(83 164 206)(84 165 207)(85 166 208)(86 167 201)(87 168 202)(88 161 203)(97 114 192)(98 115 185)(99 116 186)(100 117 187)(101 118 188)(102 119 189)(103 120 190)(104 113 191)(137 215 232)(138 216 225)(139 209 226)(140 210 227)(141 211 228)(142 212 229)(143 213 230)(144 214 231)
(9 184)(10 177)(11 178)(12 179)(13 180)(14 181)(15 182)(16 183)(17 170)(18 171)(19 172)(20 173)(21 174)(22 175)(23 176)(24 169)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 105)(32 106)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 89)(48 90)(57 150)(58 151)(59 152)(60 145)(61 146)(62 147)(63 148)(64 149)(81 204)(82 205)(83 206)(84 207)(85 208)(86 201)(87 202)(88 203)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 113)(129 224)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 223)(137 215)(138 216)(139 209)(140 210)(141 211)(142 212)(143 213)(144 214)(153 198)(154 199)(155 200)(156 193)(157 194)(158 195)(159 196)(160 197)
(1 121 80 233 231)(2 232 234 73 122)(3 123 74 235 225)(4 226 236 75 124)(5 125 76 237 227)(6 228 238 77 126)(7 127 78 239 229)(8 230 240 79 128)(9 193 30 214 95)(10 96 215 31 194)(11 195 32 216 89)(12 90 209 25 196)(13 197 26 210 91)(14 92 211 27 198)(15 199 28 212 93)(16 94 213 29 200)(17 152 103 218 207)(18 208 219 104 145)(19 146 97 220 201)(20 202 221 98 147)(21 148 99 222 203)(22 204 223 100 149)(23 150 101 224 205)(24 206 217 102 151)(33 70 164 50 189)(34 190 51 165 71)(35 72 166 52 191)(36 192 53 167 65)(37 66 168 54 185)(38 186 55 161 67)(39 68 162 56 187)(40 188 49 163 69)(41 180 160 108 140)(42 141 109 153 181)(43 182 154 110 142)(44 143 111 155 183)(45 184 156 112 144)(46 137 105 157 177)(47 178 158 106 138)(48 139 107 159 179)(57 118 129 82 176)(58 169 83 130 119)(59 120 131 84 170)(60 171 85 132 113)(61 114 133 86 172)(62 173 87 134 115)(63 116 135 88 174)(64 175 81 136 117)
(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 34 5 38)(2 33 6 37)(3 40 7 36)(4 39 8 35)(9 120 13 116)(10 119 14 115)(11 118 15 114)(12 117 16 113)(17 140 21 144)(18 139 22 143)(19 138 23 142)(20 137 24 141)(25 81 29 85)(26 88 30 84)(27 87 31 83)(28 86 32 82)(41 148 45 152)(42 147 46 151)(43 146 47 150)(44 145 48 149)(49 78 53 74)(50 77 54 73)(51 76 55 80)(52 75 56 79)(57 93 61 89)(58 92 62 96)(59 91 63 95)(60 90 64 94)(65 225 69 229)(66 232 70 228)(67 231 71 227)(68 230 72 226)(97 178 101 182)(98 177 102 181)(99 184 103 180)(100 183 104 179)(105 206 109 202)(106 205 110 201)(107 204 111 208)(108 203 112 207)(121 190 125 186)(122 189 126 185)(123 188 127 192)(124 187 128 191)(129 199 133 195)(130 198 134 194)(131 197 135 193)(132 196 136 200)(153 221 157 217)(154 220 158 224)(155 219 159 223)(156 218 160 222)(161 233 165 237)(162 240 166 236)(163 239 167 235)(164 238 168 234)(169 211 173 215)(170 210 174 214)(171 209 175 213)(172 216 176 212)

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 12A 12B 12C 15A 15B 20A ··· 20F 20G ··· 20L 24A 24B 30A 30B 60A ··· 60F order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 8 8 8 8 10 10 10 10 10 10 12 12 12 15 15 20 ··· 20 20 ··· 20 24 24 30 30 60 ··· 60 size 1 1 3 3 2 2 4 6 12 20 60 2 2 2 10 10 30 30 2 2 6 6 6 6 4 8 40 4 4 4 ··· 4 12 ··· 12 20 20 4 4 8 ··· 8

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + - + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 Q16 D10 D10 D10 C5⋊D4 C5⋊D4 S3×D4 S3×D5 C5⋊Q16 S3×Q16 C2×S3×D5 S3×C5⋊D4 S3×C5⋊Q16 kernel S3×C5⋊Q16 S3×C5⋊2C8 C15⋊Q16 C5⋊Dic12 C3×C5⋊Q16 C15⋊7Q16 S3×Dic10 C5×S3×Q8 C5⋊Q16 C5×Dic3 S3×C10 S3×Q8 C5⋊2C8 Dic10 C5×Q8 C5×S3 Dic6 C4×S3 C3×Q8 Dic3 D6 C10 Q8 S3 C5 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 4 4 1 2 4 2 2 4 2

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 1 240 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 0 0 0 0 0 239 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 31 0 0 0 0 156 181 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 219 0 0 0 0 11 219
,
 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 18 0 0 0 0 20 230

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