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G = Q16⋊D15order 480 = 25·3·5

2nd semidirect product of Q16 and D15 acting via D15/C15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Q16⋊D15
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — Q8×D15 — Q16⋊D15
 Lower central C15 — C30 — C60 — Q16⋊D15
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16⋊D15
G = < a,b,c,d | a8=c15=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 788 in 120 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3 [×2], C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×2], D5 [×2], C10, Dic3 [×2], C12, C12 [×2], D6 [×2], C15, M4(2), SD16 [×2], Q16, Q16, C2×Q8, C4○D4, Dic5 [×2], C20, C20 [×2], D10 [×2], C3⋊C8, C24, Dic6 [×2], C4×S3 [×3], D12 [×2], C3×Q8 [×2], D15 [×2], C30, C8.C22, C52C8, C40, Dic10 [×2], C4×D5 [×3], D20 [×2], C5×Q8 [×2], C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, S3×Q8, Q83S3, Dic15, Dic15, C60, C60 [×2], D30, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, Q16⋊S3, C153C8, C120, Dic30, Dic30, C4×D15, C4×D15 [×2], D60, D60, Q8×C15 [×2], Q16⋊D5, C40⋊S3, C24⋊D5, Q82D15, C157Q16, C15×Q16, Q8×D15, Q83D15, Q16⋊D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D15, C8.C22, C22×D5, S3×D4, D30 [×3], D4×D5, Q16⋊S3, C22×D15, Q16⋊D5, D4×D15, Q16⋊D15

Smallest permutation representation of Q16⋊D15
On 240 points
Generators in S240
```(1 212 43 200 24 226 54 185)(2 213 44 201 25 227 55 186)(3 214 45 202 26 228 56 187)(4 215 31 203 27 229 57 188)(5 216 32 204 28 230 58 189)(6 217 33 205 29 231 59 190)(7 218 34 206 30 232 60 191)(8 219 35 207 16 233 46 192)(9 220 36 208 17 234 47 193)(10 221 37 209 18 235 48 194)(11 222 38 210 19 236 49 195)(12 223 39 196 20 237 50 181)(13 224 40 197 21 238 51 182)(14 225 41 198 22 239 52 183)(15 211 42 199 23 240 53 184)(61 162 117 125 80 174 99 136)(62 163 118 126 81 175 100 137)(63 164 119 127 82 176 101 138)(64 165 120 128 83 177 102 139)(65 151 106 129 84 178 103 140)(66 152 107 130 85 179 104 141)(67 153 108 131 86 180 105 142)(68 154 109 132 87 166 91 143)(69 155 110 133 88 167 92 144)(70 156 111 134 89 168 93 145)(71 157 112 135 90 169 94 146)(72 158 113 121 76 170 95 147)(73 159 114 122 77 171 96 148)(74 160 115 123 78 172 97 149)(75 161 116 124 79 173 98 150)
(1 88 24 69)(2 89 25 70)(3 90 26 71)(4 76 27 72)(5 77 28 73)(6 78 29 74)(7 79 30 75)(8 80 16 61)(9 81 17 62)(10 82 18 63)(11 83 19 64)(12 84 20 65)(13 85 21 66)(14 86 22 67)(15 87 23 68)(31 113 57 95)(32 114 58 96)(33 115 59 97)(34 116 60 98)(35 117 46 99)(36 118 47 100)(37 119 48 101)(38 120 49 102)(39 106 50 103)(40 107 51 104)(41 108 52 105)(42 109 53 91)(43 110 54 92)(44 111 55 93)(45 112 56 94)(121 229 147 215)(122 230 148 216)(123 231 149 217)(124 232 150 218)(125 233 136 219)(126 234 137 220)(127 235 138 221)(128 236 139 222)(129 237 140 223)(130 238 141 224)(131 239 142 225)(132 240 143 211)(133 226 144 212)(134 227 145 213)(135 228 146 214)(151 181 178 196)(152 182 179 197)(153 183 180 198)(154 184 166 199)(155 185 167 200)(156 186 168 201)(157 187 169 202)(158 188 170 203)(159 189 171 204)(160 190 172 205)(161 191 173 206)(162 192 174 207)(163 193 175 208)(164 194 176 209)(165 195 177 210)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 110)(92 109)(93 108)(94 107)(95 106)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(121 129)(122 128)(123 127)(124 126)(130 135)(131 134)(132 133)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(151 158)(152 157)(153 156)(154 155)(159 165)(160 164)(161 163)(166 167)(168 180)(169 179)(170 178)(171 177)(172 176)(173 175)(181 203)(182 202)(183 201)(184 200)(185 199)(186 198)(187 197)(188 196)(189 210)(190 209)(191 208)(192 207)(193 206)(194 205)(195 204)(211 226)(212 240)(213 239)(214 238)(215 237)(216 236)(217 235)(218 234)(219 233)(220 232)(221 231)(222 230)(223 229)(224 228)(225 227)```

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 D15 D30 D30 C8.C22 S3×D4 D4×D5 Q16⋊S3 Q16⋊D5 D4×D15 Q16⋊D15 kernel Q16⋊D15 C40⋊S3 C24⋊D5 Q8⋊2D15 C15⋊7Q16 C15×Q16 Q8×D15 Q8⋊3D15 C5×Q16 Dic15 D30 C3×Q16 C40 C5×Q8 C24 C3×Q8 Q16 C8 Q8 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 4 4 4 8 1 1 2 2 4 4 8

Matrix representation of Q16⋊D15 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 194 94 47 147 0 0 147 47 94 194 0 0 194 94 194 94 0 0 147 47 147 47
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 52 137 95 51 0 0 104 189 190 146 0 0 95 51 189 104 0 0 190 146 137 52
,
 161 110 0 0 0 0 131 94 0 0 0 0 0 0 240 1 0 0 0 0 240 0 0 0 0 0 0 0 240 1 0 0 0 0 240 0
,
 84 93 0 0 0 0 147 157 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,194,147,194,147,0,0,94,47,94,47,0,0,47,94,194,147,0,0,147,194,94,47],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,52,104,95,190,0,0,137,189,51,146,0,0,95,190,189,137,0,0,51,146,104,52],[161,131,0,0,0,0,110,94,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[84,147,0,0,0,0,93,157,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

Q16⋊D15 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes D_{15}`
`% in TeX`

`G:=Group("Q16:D15");`
`// GroupNames label`

`G:=SmallGroup(480,883);`
`// by ID`

`G=gap.SmallGroup(480,883);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,135,100,346,185,80,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^15=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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