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## G = C3×D5×Q16order 480 = 25·3·5

### Direct product of C3, D5 and Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D5×Q16
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×Q8×D5 — C3×D5×Q16
 Lower central C5 — C10 — C20 — C3×D5×Q16
 Upper central C1 — C6 — C12 — C3×Q16

Generators and relations for C3×D5×Q16
G = < a,b,c,d,e | a3=b5=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: 368 in 120 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, Q8, Q8, D5, C10, C12, C12, C2×C6, C15, C2×C8, Q16, Q16, C2×Q8, Dic5, Dic5, C20, C20, D10, C24, C24, C2×C12, C3×Q8, C3×Q8, C3×D5, C30, C2×Q16, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C2×C24, C3×Q16, C3×Q16, C6×Q8, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C8×D5, Dic20, C5⋊Q16, C5×Q16, Q8×D5, C6×Q16, C3×C52C8, C120, C3×Dic10, C3×Dic10, D5×C12, D5×C12, Q8×C15, D5×Q16, D5×C24, C3×Dic20, C3×C5⋊Q16, C15×Q16, C3×Q8×D5, C3×D5×Q16
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, Q16, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×Q16, C22×D5, C3×Q16, C6×D4, C6×D5, D4×D5, C6×Q16, D5×C2×C6, D5×Q16, C3×D4×D5, C3×D5×Q16

Smallest permutation representation of C3×D5×Q16
On 240 points
Generators in S240
(1 87 199)(2 88 200)(3 81 193)(4 82 194)(5 83 195)(6 84 196)(7 85 197)(8 86 198)(9 71 231)(10 72 232)(11 65 225)(12 66 226)(13 67 227)(14 68 228)(15 69 229)(16 70 230)(17 218 115)(18 219 116)(19 220 117)(20 221 118)(21 222 119)(22 223 120)(23 224 113)(24 217 114)(25 44 103)(26 45 104)(27 46 97)(28 47 98)(29 48 99)(30 41 100)(31 42 101)(32 43 102)(33 95 111)(34 96 112)(35 89 105)(36 90 106)(37 91 107)(38 92 108)(39 93 109)(40 94 110)(49 180 127)(50 181 128)(51 182 121)(52 183 122)(53 184 123)(54 177 124)(55 178 125)(56 179 126)(57 188 132)(58 189 133)(59 190 134)(60 191 135)(61 192 136)(62 185 129)(63 186 130)(64 187 131)(73 141 240)(74 142 233)(75 143 234)(76 144 235)(77 137 236)(78 138 237)(79 139 238)(80 140 239)(145 161 203)(146 162 204)(147 163 205)(148 164 206)(149 165 207)(150 166 208)(151 167 201)(152 168 202)(153 169 214)(154 170 215)(155 171 216)(156 172 209)(157 173 210)(158 174 211)(159 175 212)(160 176 213)
(1 63 51 78 229)(2 64 52 79 230)(3 57 53 80 231)(4 58 54 73 232)(5 59 55 74 225)(6 60 56 75 226)(7 61 49 76 227)(8 62 50 77 228)(9 81 188 184 140)(10 82 189 177 141)(11 83 190 178 142)(12 84 191 179 143)(13 85 192 180 144)(14 86 185 181 137)(15 87 186 182 138)(16 88 187 183 139)(17 92 32 153 148)(18 93 25 154 149)(19 94 26 155 150)(20 95 27 156 151)(21 96 28 157 152)(22 89 29 158 145)(23 90 30 159 146)(24 91 31 160 147)(33 97 209 201 118)(34 98 210 202 119)(35 99 211 203 120)(36 100 212 204 113)(37 101 213 205 114)(38 102 214 206 115)(39 103 215 207 116)(40 104 216 208 117)(41 175 162 224 106)(42 176 163 217 107)(43 169 164 218 108)(44 170 165 219 109)(45 171 166 220 110)(46 172 167 221 111)(47 173 168 222 112)(48 174 161 223 105)(65 195 134 125 233)(66 196 135 126 234)(67 197 136 127 235)(68 198 129 128 236)(69 199 130 121 237)(70 200 131 122 238)(71 193 132 123 239)(72 194 133 124 240)
(1 229)(2 230)(3 231)(4 232)(5 225)(6 226)(7 227)(8 228)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 105)(57 80)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 195)(66 196)(67 197)(68 198)(69 199)(70 200)(71 193)(72 194)(113 212)(114 213)(115 214)(116 215)(117 216)(118 209)(119 210)(120 211)(129 236)(130 237)(131 238)(132 239)(133 240)(134 233)(135 234)(136 235)(137 185)(138 186)(139 187)(140 188)(141 189)(142 190)(143 191)(144 192)(169 218)(170 219)(171 220)(172 221)(173 222)(174 223)(175 224)(176 217)
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 107 13 111)(10 106 14 110)(11 105 15 109)(12 112 16 108)(17 75 21 79)(18 74 22 78)(19 73 23 77)(20 80 24 76)(33 71 37 67)(34 70 38 66)(35 69 39 65)(36 68 40 72)(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)(57 160 61 156)(58 159 62 155)(59 158 63 154)(60 157 64 153)(89 229 93 225)(90 228 94 232)(91 227 95 231)(92 226 96 230)(97 193 101 197)(98 200 102 196)(99 199 103 195)(100 198 104 194)(113 236 117 240)(114 235 118 239)(115 234 119 238)(116 233 120 237)(121 207 125 203)(122 206 126 202)(123 205 127 201)(124 204 128 208)(129 216 133 212)(130 215 134 211)(131 214 135 210)(132 213 136 209)(137 220 141 224)(138 219 142 223)(139 218 143 222)(140 217 144 221)(161 182 165 178)(162 181 166 177)(163 180 167 184)(164 179 168 183)(169 191 173 187)(170 190 174 186)(171 189 175 185)(172 188 176 192)

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

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

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

84 conjugacy classes

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

84 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 C3 C6 C6 C6 C6 C6 D4 D4 D5 Q16 D10 D10 C3×D4 C3×D4 C3×D5 C3×Q16 C6×D5 C6×D5 D4×D5 D5×Q16 C3×D4×D5 C3×D5×Q16 kernel C3×D5×Q16 D5×C24 C3×Dic20 C3×C5⋊Q16 C15×Q16 C3×Q8×D5 D5×Q16 C8×D5 Dic20 C5⋊Q16 C5×Q16 Q8×D5 C3×Dic5 C6×D5 C3×Q16 C3×D5 C24 C3×Q8 Dic5 D10 Q16 D5 C8 Q8 C6 C3 C2 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 2 4 2 4 2 2 4 8 4 8 2 4 4 8

Matrix representation of C3×D5×Q16 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 15 0 0 0 0 15
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 240 189
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 0 57 0 0 93 22 0 0 0 0 1 0 0 0 0 1
,
 202 49 0 0 146 39 0 0 0 0 240 0 0 0 0 240
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,189],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,93,0,0,57,22,0,0,0,0,1,0,0,0,0,1],[202,146,0,0,49,39,0,0,0,0,240,0,0,0,0,240] >;

C3×D5×Q16 in GAP, Magma, Sage, TeX

C_3\times D_5\times Q_{16}
% in TeX

G:=Group("C3xD5xQ16");
// GroupNames label

G:=SmallGroup(480,710);
// by ID

G=gap.SmallGroup(480,710);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,303,268,1271,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^5=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

׿
×
𝔽