D15:Q16 - GroupNames

G = D₁₅⋊Q₁₆ order 480 = 2⁵·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₅⋊₂Q₁₆, D₃₀.₄₂D₄, C₆₀.₃₅C₂³, Dic₁₀.₁₂D₆, Dic₆.₁₂D₁₀, Dic₁₅.₁₆D₄, Dic₃₀.₁₂C₂², C₅⋊₃(S₃×Q₁₆), C₃⋊₃(D₅×Q₁₆), C₁₅⋊₉(C₂×Q₁₆), C₃⋊C₈.₁₈D₁₀, C₃⋊Q₁₆⋊₂D₅, C₅⋊Q₁₆⋊₂S₃, C₆.₇₇(D₄×D₅), C₃⋊Dic₂₀⋊₆C₂, C₅⋊Dic₁₂⋊₆C₂, C₁₀.₇₈(S₃×D₄), C₅⋊₂C₈.₁₈D₆, D₁₅⋊Q₈.₁C₂, Q₈.₂₀(S₃×D₅), (Q₈×D₁₅).₁C₂, (C₃×Q₈).₃D₁₀, (C₅×Q₈).₂₀D₆, C₃₀.₁₉₇(C₂×D₄), D₁₅⋊₂C₈.₁C₂, C₂₀.₃₅(C₂²×S₃), C₁₂.₃₅(C₂²×D₅), (Q₈×C₁₅).₅C₂², (C₄×D₁₅).₁₁C₂², C₂.₃₀(D₁₀⋊D₆), (C₅×Dic₆).₁₁C₂², (C₃×Dic₁₀).₁₁C₂², C₄.₃₅(C₂×S₃×D₅), (C₃×C₅⋊Q₁₆)⋊₃C₂, (C₅×C₃⋊Q₁₆)⋊₃C₂, (C₅×C₃⋊C₈).₉C₂², (C₃×C₅⋊₂C₈).₉C₂², SmallGroup(480,587)

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, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₈, C₂×C₄, Q₈, Q₈, D₅, C₁₀, Dic₃, C₁₂, C₁₂, D₆, C₁₅, C₂×C₈, Q₁₆, C₂×Q₈, Dic₅, C₂₀, C₂₀, D₁₀, C₃⋊C₈, C₂₄, Dic₆, Dic₆, C₄×S₃, C₃×Q₈, C₃×Q₈, D₁₅, C₃₀, C₂×Q₁₆, C₅⋊₂C₈, C₄₀, Dic₁₀, Dic₁₀, C₄×D₅, C₅×Q₈, C₅×Q₈, S₃×C₈, Dic₁₂, C₃⋊Q₁₆, C₃⋊Q₁₆, C₃×Q₁₆, S₃×Q₈, C₅×Dic₃, C₃×Dic₅, Dic₁₅, Dic₁₅, C₆₀, C₆₀, D₃₀, C₈×D₅, Dic₂₀, C₅⋊Q₁₆, C₅⋊Q₁₆, C₅×Q₁₆, Q₈×D₅, S₃×Q₁₆, C₅×C₃⋊C₈, C₃×C₅⋊₂C₈, D₃₀.C₂, C₁₅⋊Q₈, C₃×Dic₁₀, C₅×Dic₆, Dic₃₀, Dic₃₀, C₄×D₁₅, C₄×D₁₅, Q₈×C₁₅, D₅×Q₁₆, D₁₅⋊₂C₈, C₃⋊Dic₂₀, C₅⋊Dic₁₂, C₃×C₅⋊Q₁₆, C₅×C₃⋊Q₁₆, D₁₅⋊Q₈, Q₈×D₁₅, D₁₅⋊Q₁₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, Q₁₆, C₂×D₄, D₁₀, 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)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 68)(62 67)(63 66)(64 65)(69 75)(70 74)(71 73)(76 83)(77 82)(78 81)(79 80)(84 90)(85 89)(86 88)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(106 107)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(136 140)(137 139)(141 150)(142 149)(143 148)(144 147)(145 146)(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)(181 186)(182 185)(183 184)(187 195)(188 194)(189 193)(190 192)(196 205)(197 204)(198 203)(199 202)(200 201)(206 210)(207 209)(211 222)(212 221)(213 220)(214 219)(215 218)(216 217)(223 225)(226 231)(227 230)(228 229)(232 240)(233 239)(234 238)(235 237)

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

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

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

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

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

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

G = D15⋊Q16 order 480 = 25·3·5

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

G = D₁₅⋊Q₁₆ order 480 = 2⁵·3·5

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