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## G = D5×C3⋊Q16order 480 = 25·3·5

### Direct product of D5 and C3⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D5×C3⋊Q16
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×Dic6 — D5×C3⋊Q16
 Lower central C15 — C30 — C60 — D5×C3⋊Q16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D5×C3⋊Q16
G = < a,b,c,d,e | a5=b2=c3=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: 556 in 120 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, Q8, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, Dic5, Dic5, C20, C20, D10, C3⋊C8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C3×D5, C30, C2×Q16, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C5×Q8, C2×C3⋊C8, C3⋊Q16, C3⋊Q16, C2×Dic6, C6×Q8, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C60, C6×D5, C8×D5, Dic20, C5⋊Q16, C5×Q16, Q8×D5, Q8×D5, C2×C3⋊Q16, C5×C3⋊C8, C153C8, D5×Dic3, C15⋊Q8, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C5×Dic6, Dic30, Q8×C15, D5×Q16, D5×C3⋊C8, C15⋊Q16, C3⋊Dic20, C5×C3⋊Q16, C157Q16, D5×Dic6, C3×Q8×D5, D5×C3⋊Q16
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, Q16, C2×D4, D10, C3⋊D4, C22×S3, C2×Q16, C22×D5, C3⋊Q16, C2×C3⋊D4, S3×D5, D4×D5, C2×C3⋊Q16, C2×S3×D5, D5×Q16, D5×C3⋊D4, D5×C3⋊Q16

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

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

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

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

48 conjugacy classes

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

48 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 C3⋊D4 C3⋊D4 C3⋊Q16 S3×D5 D4×D5 C2×S3×D5 D5×Q16 D5×C3⋊D4 D5×C3⋊Q16 kernel D5×C3⋊Q16 D5×C3⋊C8 C15⋊Q16 C3⋊Dic20 C5×C3⋊Q16 C15⋊7Q16 D5×Dic6 C3×Q8×D5 Q8×D5 C3×Dic5 C6×D5 C3⋊Q16 Dic10 C4×D5 C5×Q8 C3×D5 C3⋊C8 Dic6 C3×Q8 Dic5 D10 D5 Q8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 2 2 2 2 2 2 4 4 2

Matrix representation of D5×C3⋊Q16 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 189 1 0 0 0 0 240 0 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 1 189 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 240 0 0 0 0 1 240 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
,
 166 190 0 0 0 0 115 75 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 11 11 0 0 0 0 230 11
,
 1 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 0 222 222 0 0 0 0 222 19

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,189,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[166,115,0,0,0,0,190,75,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,11,230,0,0,0,0,11,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,222,222,0,0,0,0,222,19] >;

D5×C3⋊Q16 in GAP, Magma, Sage, TeX

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

G:=Group("D5xC3:Q16");
// GroupNames label

G:=SmallGroup(480,583);
// by ID

G=gap.SmallGroup(480,583);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,135,100,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^3=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

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