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## G = D15⋊Q16order 480 = 25·3·5

### The semidirect product of D15 and Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D15⋊Q16
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — D15⋊Q8 — D15⋊Q16
 Lower central C15 — C30 — C60 — D15⋊Q16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D15⋊Q16
G = < a,b,c,d | a15=b2=c8=1, d2=c4, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, bd=db, dcd-1=c-1 >

Subgroups: 652 in 120 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C5, S3 [×2], C6, C8 [×2], C2×C4 [×3], Q8, Q8 [×5], D5 [×2], C10, Dic3 [×3], C12, C12 [×2], D6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×3], C20, C20 [×2], D10, C3⋊C8, C24, Dic6, Dic6 [×3], C4×S3 [×3], C3×Q8, C3×Q8, D15 [×2], C30, C2×Q16, C52C8, C40, Dic10, Dic10 [×3], C4×D5 [×3], C5×Q8, C5×Q8, S3×C8, Dic12, C3⋊Q16, C3⋊Q16, C3×Q16, S3×Q8 [×2], C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C60, D30, C8×D5, Dic20, C5⋊Q16, C5⋊Q16, C5×Q16, Q8×D5 [×2], S3×Q16, C5×C3⋊C8, C3×C52C8, D30.C2, C15⋊Q8, C3×Dic10, C5×Dic6, Dic30, Dic30, C4×D15, C4×D15, Q8×C15, D5×Q16, D152C8, C3⋊Dic20, C5⋊Dic12, C3×C5⋊Q16, C5×C3⋊Q16, D15⋊Q8, Q8×D15, D15⋊Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], C22×S3, C2×Q16, C22×D5, S3×D4, S3×D5, D4×D5, S3×Q16, C2×S3×D5, D5×Q16, D10⋊D6, D15⋊Q16

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

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

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

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

45 conjugacy classes

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

45 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 8 type + + + + + + + + + + + + + + + - + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 Q16 D10 D10 D10 S3×D4 S3×D5 D4×D5 S3×Q16 C2×S3×D5 D5×Q16 D10⋊D6 D15⋊Q16 kernel D15⋊Q16 D15⋊2C8 C3⋊Dic20 C5⋊Dic12 C3×C5⋊Q16 C5×C3⋊Q16 D15⋊Q8 Q8×D15 C5⋊Q16 Dic15 D30 C3⋊Q16 C5⋊2C8 Dic10 C5×Q8 D15 C3⋊C8 Dic6 C3×Q8 C10 Q8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 1 2 2 2 2 4 4 2

Matrix representation of D15⋊Q16 in GL6(𝔽241)

 240 189 0 0 0 0 52 52 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
,
 240 189 0 0 0 0 0 1 0 0 0 0 0 0 1 240 0 0 0 0 0 240 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 0 230 230 0 0 0 0 11 230
,
 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 0 113 56 0 0 0 0 56 128

`G:=sub<GL(6,GF(241))| [240,52,0,0,0,0,189,52,0,0,0,0,0,0,0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,189,1,0,0,0,0,0,0,1,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,230,11,0,0,0,0,230,230],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,113,56,0,0,0,0,56,128] >;`

D15⋊Q16 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(480,587);`
`// by ID`

`G=gap.SmallGroup(480,587);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,135,100,675,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^15=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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