metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.3D30, Q16⋊2D15, C40.30D6, Q8.9D30, D30.33D4, C24.30D10, C60.72C23, C120.27C22, Dic15.38D4, D60.23C22, Dic30.26C22, (C3×Q16)⋊4D5, (C5×Q16)⋊4S3, C40⋊S3⋊4C2, C24⋊D5⋊6C2, (C15×Q16)⋊4C2, (Q8×D15)⋊10C2, C6.116(D4×D5), C2.23(D4×D15), C5⋊5(Q16⋊S3), (C5×Q8).33D6, C3⋊5(Q16⋊D5), C15⋊7Q16⋊12C2, C10.118(S3×D4), C30.323(C2×D4), (C3×Q8).16D10, C4.9(C22×D15), Q8⋊2D15⋊11C2, Q8⋊3D15.2C2, C15⋊31(C8.C22), C20.110(C22×S3), C15⋊3C8.20C22, (C4×D15).27C22, C12.110(C22×D5), (Q8×C15).25C22, SmallGroup(480,883)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C5, S3, C6, C8, C8, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, C12, D6, C15, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C3⋊C8, C24, Dic6, C4×S3, D12, C3×Q8, D15, C30, C8.C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C5×Q8, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, Dic15, Dic15, C60, C60, D30, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, Q16⋊S3, C15⋊3C8, C120, Dic30, Dic30, C4×D15, C4×D15, D60, D60, Q8×C15, Q16⋊D5, C40⋊S3, C24⋊D5, Q8⋊2D15, C15⋊7Q16, C15×Q16, Q8×D15, Q8⋊3D15, Q16⋊D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, D15, C8.C22, C22×D5, S3×D4, D30, D4×D5, Q16⋊S3, C22×D15, Q16⋊D5, D4×D15, Q16⋊D15
►On 240 points
Generators in S240
(1 211 35 208 20 227 46 182)(2 212 36 209 21 228 47 183)(3 213 37 210 22 229 48 184)(4 214 38 196 23 230 49 185)(5 215 39 197 24 231 50 186)(6 216 40 198 25 232 51 187)(7 217 41 199 26 233 52 188)(8 218 42 200 27 234 53 189)(9 219 43 201 28 235 54 190)(10 220 44 202 29 236 55 191)(11 221 45 203 30 237 56 192)(12 222 31 204 16 238 57 193)(13 223 32 205 17 239 58 194)(14 224 33 206 18 240 59 195)(15 225 34 207 19 226 60 181)(61 151 117 125 87 177 99 140)(62 152 118 126 88 178 100 141)(63 153 119 127 89 179 101 142)(64 154 120 128 90 180 102 143)(65 155 106 129 76 166 103 144)(66 156 107 130 77 167 104 145)(67 157 108 131 78 168 105 146)(68 158 109 132 79 169 91 147)(69 159 110 133 80 170 92 148)(70 160 111 134 81 171 93 149)(71 161 112 135 82 172 94 150)(72 162 113 121 83 173 95 136)(73 163 114 122 84 174 96 137)(74 164 115 123 85 175 97 138)(75 165 116 124 86 176 98 139)
(1 76 20 65)(2 77 21 66)(3 78 22 67)(4 79 23 68)(5 80 24 69)(6 81 25 70)(7 82 26 71)(8 83 27 72)(9 84 28 73)(10 85 29 74)(11 86 30 75)(12 87 16 61)(13 88 17 62)(14 89 18 63)(15 90 19 64)(31 117 57 99)(32 118 58 100)(33 119 59 101)(34 120 60 102)(35 106 46 103)(36 107 47 104)(37 108 48 105)(38 109 49 91)(39 110 50 92)(40 111 51 93)(41 112 52 94)(42 113 53 95)(43 114 54 96)(44 115 55 97)(45 116 56 98)(121 234 136 218)(122 235 137 219)(123 236 138 220)(124 237 139 221)(125 238 140 222)(126 239 141 223)(127 240 142 224)(128 226 143 225)(129 227 144 211)(130 228 145 212)(131 229 146 213)(132 230 147 214)(133 231 148 215)(134 232 149 216)(135 233 150 217)(151 193 177 204)(152 194 178 205)(153 195 179 206)(154 181 180 207)(155 182 166 208)(156 183 167 209)(157 184 168 210)(158 185 169 196)(159 186 170 197)(160 187 171 198)(161 188 172 199)(162 189 173 200)(163 190 174 201)(164 191 175 202)(165 192 176 203)
(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)(16 23)(17 22)(18 21)(19 20)(24 30)(25 29)(26 28)(31 38)(32 37)(33 36)(34 35)(39 45)(40 44)(41 43)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 79)(62 78)(63 77)(64 76)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 120)(104 119)(105 118)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(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)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(181 208)(182 207)(183 206)(184 205)(185 204)(186 203)(187 202)(188 201)(189 200)(190 199)(191 198)(192 197)(193 196)(194 210)(195 209)(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)| (1,211,35,208,20,227,46,182)(2,212,36,209,21,228,47,183)(3,213,37,210,22,229,48,184)(4,214,38,196,23,230,49,185)(5,215,39,197,24,231,50,186)(6,216,40,198,25,232,51,187)(7,217,41,199,26,233,52,188)(8,218,42,200,27,234,53,189)(9,219,43,201,28,235,54,190)(10,220,44,202,29,236,55,191)(11,221,45,203,30,237,56,192)(12,222,31,204,16,238,57,193)(13,223,32,205,17,239,58,194)(14,224,33,206,18,240,59,195)(15,225,34,207,19,226,60,181)(61,151,117,125,87,177,99,140)(62,152,118,126,88,178,100,141)(63,153,119,127,89,179,101,142)(64,154,120,128,90,180,102,143)(65,155,106,129,76,166,103,144)(66,156,107,130,77,167,104,145)(67,157,108,131,78,168,105,146)(68,158,109,132,79,169,91,147)(69,159,110,133,80,170,92,148)(70,160,111,134,81,171,93,149)(71,161,112,135,82,172,94,150)(72,162,113,121,83,173,95,136)(73,163,114,122,84,174,96,137)(74,164,115,123,85,175,97,138)(75,165,116,124,86,176,98,139), (1,76,20,65)(2,77,21,66)(3,78,22,67)(4,79,23,68)(5,80,24,69)(6,81,25,70)(7,82,26,71)(8,83,27,72)(9,84,28,73)(10,85,29,74)(11,86,30,75)(12,87,16,61)(13,88,17,62)(14,89,18,63)(15,90,19,64)(31,117,57,99)(32,118,58,100)(33,119,59,101)(34,120,60,102)(35,106,46,103)(36,107,47,104)(37,108,48,105)(38,109,49,91)(39,110,50,92)(40,111,51,93)(41,112,52,94)(42,113,53,95)(43,114,54,96)(44,115,55,97)(45,116,56,98)(121,234,136,218)(122,235,137,219)(123,236,138,220)(124,237,139,221)(125,238,140,222)(126,239,141,223)(127,240,142,224)(128,226,143,225)(129,227,144,211)(130,228,145,212)(131,229,146,213)(132,230,147,214)(133,231,148,215)(134,232,149,216)(135,233,150,217)(151,193,177,204)(152,194,178,205)(153,195,179,206)(154,181,180,207)(155,182,166,208)(156,183,167,209)(157,184,168,210)(158,185,169,196)(159,186,170,197)(160,187,171,198)(161,188,172,199)(162,189,173,200)(163,190,174,201)(164,191,175,202)(165,192,176,203), (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)(16,23)(17,22)(18,21)(19,20)(24,30)(25,29)(26,28)(31,38)(32,37)(33,36)(34,35)(39,45)(40,44)(41,43)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,79)(62,78)(63,77)(64,76)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,120)(104,119)(105,118)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(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)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,208)(182,207)(183,206)(184,205)(185,204)(186,203)(187,202)(188,201)(189,200)(190,199)(191,198)(192,197)(193,196)(194,210)(195,209)(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( (1,211,35,208,20,227,46,182)(2,212,36,209,21,228,47,183)(3,213,37,210,22,229,48,184)(4,214,38,196,23,230,49,185)(5,215,39,197,24,231,50,186)(6,216,40,198,25,232,51,187)(7,217,41,199,26,233,52,188)(8,218,42,200,27,234,53,189)(9,219,43,201,28,235,54,190)(10,220,44,202,29,236,55,191)(11,221,45,203,30,237,56,192)(12,222,31,204,16,238,57,193)(13,223,32,205,17,239,58,194)(14,224,33,206,18,240,59,195)(15,225,34,207,19,226,60,181)(61,151,117,125,87,177,99,140)(62,152,118,126,88,178,100,141)(63,153,119,127,89,179,101,142)(64,154,120,128,90,180,102,143)(65,155,106,129,76,166,103,144)(66,156,107,130,77,167,104,145)(67,157,108,131,78,168,105,146)(68,158,109,132,79,169,91,147)(69,159,110,133,80,170,92,148)(70,160,111,134,81,171,93,149)(71,161,112,135,82,172,94,150)(72,162,113,121,83,173,95,136)(73,163,114,122,84,174,96,137)(74,164,115,123,85,175,97,138)(75,165,116,124,86,176,98,139), (1,76,20,65)(2,77,21,66)(3,78,22,67)(4,79,23,68)(5,80,24,69)(6,81,25,70)(7,82,26,71)(8,83,27,72)(9,84,28,73)(10,85,29,74)(11,86,30,75)(12,87,16,61)(13,88,17,62)(14,89,18,63)(15,90,19,64)(31,117,57,99)(32,118,58,100)(33,119,59,101)(34,120,60,102)(35,106,46,103)(36,107,47,104)(37,108,48,105)(38,109,49,91)(39,110,50,92)(40,111,51,93)(41,112,52,94)(42,113,53,95)(43,114,54,96)(44,115,55,97)(45,116,56,98)(121,234,136,218)(122,235,137,219)(123,236,138,220)(124,237,139,221)(125,238,140,222)(126,239,141,223)(127,240,142,224)(128,226,143,225)(129,227,144,211)(130,228,145,212)(131,229,146,213)(132,230,147,214)(133,231,148,215)(134,232,149,216)(135,233,150,217)(151,193,177,204)(152,194,178,205)(153,195,179,206)(154,181,180,207)(155,182,166,208)(156,183,167,209)(157,184,168,210)(158,185,169,196)(159,186,170,197)(160,187,171,198)(161,188,172,199)(162,189,173,200)(163,190,174,201)(164,191,175,202)(165,192,176,203), (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)(16,23)(17,22)(18,21)(19,20)(24,30)(25,29)(26,28)(31,38)(32,37)(33,36)(34,35)(39,45)(40,44)(41,43)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,79)(62,78)(63,77)(64,76)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,120)(104,119)(105,118)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(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)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,208)(182,207)(183,206)(184,205)(185,204)(186,203)(187,202)(188,201)(189,200)(190,199)(191,198)(192,197)(193,196)(194,210)(195,209)(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([[(1,211,35,208,20,227,46,182),(2,212,36,209,21,228,47,183),(3,213,37,210,22,229,48,184),(4,214,38,196,23,230,49,185),(5,215,39,197,24,231,50,186),(6,216,40,198,25,232,51,187),(7,217,41,199,26,233,52,188),(8,218,42,200,27,234,53,189),(9,219,43,201,28,235,54,190),(10,220,44,202,29,236,55,191),(11,221,45,203,30,237,56,192),(12,222,31,204,16,238,57,193),(13,223,32,205,17,239,58,194),(14,224,33,206,18,240,59,195),(15,225,34,207,19,226,60,181),(61,151,117,125,87,177,99,140),(62,152,118,126,88,178,100,141),(63,153,119,127,89,179,101,142),(64,154,120,128,90,180,102,143),(65,155,106,129,76,166,103,144),(66,156,107,130,77,167,104,145),(67,157,108,131,78,168,105,146),(68,158,109,132,79,169,91,147),(69,159,110,133,80,170,92,148),(70,160,111,134,81,171,93,149),(71,161,112,135,82,172,94,150),(72,162,113,121,83,173,95,136),(73,163,114,122,84,174,96,137),(74,164,115,123,85,175,97,138),(75,165,116,124,86,176,98,139)], [(1,76,20,65),(2,77,21,66),(3,78,22,67),(4,79,23,68),(5,80,24,69),(6,81,25,70),(7,82,26,71),(8,83,27,72),(9,84,28,73),(10,85,29,74),(11,86,30,75),(12,87,16,61),(13,88,17,62),(14,89,18,63),(15,90,19,64),(31,117,57,99),(32,118,58,100),(33,119,59,101),(34,120,60,102),(35,106,46,103),(36,107,47,104),(37,108,48,105),(38,109,49,91),(39,110,50,92),(40,111,51,93),(41,112,52,94),(42,113,53,95),(43,114,54,96),(44,115,55,97),(45,116,56,98),(121,234,136,218),(122,235,137,219),(123,236,138,220),(124,237,139,221),(125,238,140,222),(126,239,141,223),(127,240,142,224),(128,226,143,225),(129,227,144,211),(130,228,145,212),(131,229,146,213),(132,230,147,214),(133,231,148,215),(134,232,149,216),(135,233,150,217),(151,193,177,204),(152,194,178,205),(153,195,179,206),(154,181,180,207),(155,182,166,208),(156,183,167,209),(157,184,168,210),(158,185,169,196),(159,186,170,197),(160,187,171,198),(161,188,172,199),(162,189,173,200),(163,190,174,201),(164,191,175,202),(165,192,176,203)], [(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),(16,23),(17,22),(18,21),(19,20),(24,30),(25,29),(26,28),(31,38),(32,37),(33,36),(34,35),(39,45),(40,44),(41,43),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,79),(62,78),(63,77),(64,76),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,84),(72,83),(73,82),(74,81),(75,80),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,120),(104,119),(105,118),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129),(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),(166,180),(167,179),(168,178),(169,177),(170,176),(171,175),(172,174),(181,208),(182,207),(183,206),(184,205),(185,204),(186,203),(187,202),(188,201),(189,200),(190,199),(191,198),(192,197),(193,196),(194,210),(195,209),(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