direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16×D15, C8.9D30, C40.23D6, Q8.8D30, Dic60⋊8C2, D30.50D4, C24.23D10, C60.71C23, C120.14C22, Dic15.24D4, Dic30.25C22, C3⋊4(D5×Q16), C5⋊4(S3×Q16), (C5×Q16)⋊2S3, (C3×Q16)⋊2D5, C15⋊13(C2×Q16), (C15×Q16)⋊2C2, (C8×D15).1C2, C6.115(D4×D5), C2.22(D4×D15), (Q8×D15).2C2, (C5×Q8).32D6, C15⋊7Q16⋊11C2, C30.322(C2×D4), C10.117(S3×D4), (C3×Q8).15D10, C4.8(C22×D15), C20.109(C22×S3), C15⋊3C8.33C22, (C4×D15).46C22, C12.109(C22×D5), (Q8×C15).24C22, SmallGroup(480,882)
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, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 692 in 120 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C8, C2×C4, Q8, Q8, D5, C10, Dic3, C12, C12, D6, C15, C2×C8, Q16, Q16, C2×Q8, Dic5, C20, C20, D10, C3⋊C8, C24, Dic6, C4×S3, C3×Q8, D15, C30, C2×Q16, C5⋊2C8, C40, Dic10, C4×D5, C5×Q8, S3×C8, Dic12, C3⋊Q16, C3×Q16, S3×Q8, Dic15, Dic15, C60, C60, D30, C8×D5, Dic20, C5⋊Q16, C5×Q16, Q8×D5, S3×Q16, C15⋊3C8, C120, Dic30, Dic30, C4×D15, C4×D15, Q8×C15, D5×Q16, C8×D15, Dic60, C15⋊7Q16, C15×Q16, Q8×D15, Q16×D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, Q16, C2×D4, D10, C22×S3, D15, C2×Q16, C22×D5, S3×D4, D30, D4×D5, S3×Q16, C22×D15, D5×Q16, D4×D15, Q16×D15
►On 240 points
Generators in S240
(1 219 32 200 20 238 58 189)(2 220 33 201 21 239 59 190)(3 221 34 202 22 240 60 191)(4 222 35 203 23 226 46 192)(5 223 36 204 24 227 47 193)(6 224 37 205 25 228 48 194)(7 225 38 206 26 229 49 195)(8 211 39 207 27 230 50 181)(9 212 40 208 28 231 51 182)(10 213 41 209 29 232 52 183)(11 214 42 210 30 233 53 184)(12 215 43 196 16 234 54 185)(13 216 44 197 17 235 55 186)(14 217 45 198 18 236 56 187)(15 218 31 199 19 237 57 188)(61 154 110 121 80 173 91 143)(62 155 111 122 81 174 92 144)(63 156 112 123 82 175 93 145)(64 157 113 124 83 176 94 146)(65 158 114 125 84 177 95 147)(66 159 115 126 85 178 96 148)(67 160 116 127 86 179 97 149)(68 161 117 128 87 180 98 150)(69 162 118 129 88 166 99 136)(70 163 119 130 89 167 100 137)(71 164 120 131 90 168 101 138)(72 165 106 132 76 169 102 139)(73 151 107 133 77 170 103 140)(74 152 108 134 78 171 104 141)(75 153 109 135 79 172 105 142)
(1 77 20 73)(2 78 21 74)(3 79 22 75)(4 80 23 61)(5 81 24 62)(6 82 25 63)(7 83 26 64)(8 84 27 65)(9 85 28 66)(10 86 29 67)(11 87 30 68)(12 88 16 69)(13 89 17 70)(14 90 18 71)(15 76 19 72)(31 106 57 102)(32 107 58 103)(33 108 59 104)(34 109 60 105)(35 110 46 91)(36 111 47 92)(37 112 48 93)(38 113 49 94)(39 114 50 95)(40 115 51 96)(41 116 52 97)(42 117 53 98)(43 118 54 99)(44 119 55 100)(45 120 56 101)(121 226 143 222)(122 227 144 223)(123 228 145 224)(124 229 146 225)(125 230 147 211)(126 231 148 212)(127 232 149 213)(128 233 150 214)(129 234 136 215)(130 235 137 216)(131 236 138 217)(132 237 139 218)(133 238 140 219)(134 239 141 220)(135 240 142 221)(151 189 170 200)(152 190 171 201)(153 191 172 202)(154 192 173 203)(155 193 174 204)(156 194 175 205)(157 195 176 206)(158 181 177 207)(159 182 178 208)(160 183 179 209)(161 184 180 210)(162 185 166 196)(163 186 167 197)(164 187 168 198)(165 188 169 199)
(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 19)(2 18)(3 17)(4 16)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 60)(45 59)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 90)(75 89)(91 118)(92 117)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 120)(105 119)(121 136)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)(134 138)(135 137)(151 169)(152 168)(153 167)(154 166)(155 180)(156 179)(157 178)(158 177)(159 176)(160 175)(161 174)(162 173)(163 172)(164 171)(165 170)(181 207)(182 206)(183 205)(184 204)(185 203)(186 202)(187 201)(188 200)(189 199)(190 198)(191 197)(192 196)(193 210)(194 209)(195 208)(211 230)(212 229)(213 228)(214 227)(215 226)(216 240)(217 239)(218 238)(219 237)(220 236)(221 235)(222 234)(223 233)(224 232)(225 231)
G:=sub<Sym(240)| (1,219,32,200,20,238,58,189)(2,220,33,201,21,239,59,190)(3,221,34,202,22,240,60,191)(4,222,35,203,23,226,46,192)(5,223,36,204,24,227,47,193)(6,224,37,205,25,228,48,194)(7,225,38,206,26,229,49,195)(8,211,39,207,27,230,50,181)(9,212,40,208,28,231,51,182)(10,213,41,209,29,232,52,183)(11,214,42,210,30,233,53,184)(12,215,43,196,16,234,54,185)(13,216,44,197,17,235,55,186)(14,217,45,198,18,236,56,187)(15,218,31,199,19,237,57,188)(61,154,110,121,80,173,91,143)(62,155,111,122,81,174,92,144)(63,156,112,123,82,175,93,145)(64,157,113,124,83,176,94,146)(65,158,114,125,84,177,95,147)(66,159,115,126,85,178,96,148)(67,160,116,127,86,179,97,149)(68,161,117,128,87,180,98,150)(69,162,118,129,88,166,99,136)(70,163,119,130,89,167,100,137)(71,164,120,131,90,168,101,138)(72,165,106,132,76,169,102,139)(73,151,107,133,77,170,103,140)(74,152,108,134,78,171,104,141)(75,153,109,135,79,172,105,142), (1,77,20,73)(2,78,21,74)(3,79,22,75)(4,80,23,61)(5,81,24,62)(6,82,25,63)(7,83,26,64)(8,84,27,65)(9,85,28,66)(10,86,29,67)(11,87,30,68)(12,88,16,69)(13,89,17,70)(14,90,18,71)(15,76,19,72)(31,106,57,102)(32,107,58,103)(33,108,59,104)(34,109,60,105)(35,110,46,91)(36,111,47,92)(37,112,48,93)(38,113,49,94)(39,114,50,95)(40,115,51,96)(41,116,52,97)(42,117,53,98)(43,118,54,99)(44,119,55,100)(45,120,56,101)(121,226,143,222)(122,227,144,223)(123,228,145,224)(124,229,146,225)(125,230,147,211)(126,231,148,212)(127,232,149,213)(128,233,150,214)(129,234,136,215)(130,235,137,216)(131,236,138,217)(132,237,139,218)(133,238,140,219)(134,239,141,220)(135,240,142,221)(151,189,170,200)(152,190,171,201)(153,191,172,202)(154,192,173,203)(155,193,174,204)(156,194,175,205)(157,195,176,206)(158,181,177,207)(159,182,178,208)(160,183,179,209)(161,184,180,210)(162,185,166,196)(163,186,167,197)(164,187,168,198)(165,188,169,199), (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), (1,19)(2,18)(3,17)(4,16)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,60)(45,59)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,90)(75,89)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,120)(105,119)(121,136)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)(151,169)(152,168)(153,167)(154,166)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(181,207)(182,206)(183,205)(184,204)(185,203)(186,202)(187,201)(188,200)(189,199)(190,198)(191,197)(192,196)(193,210)(194,209)(195,208)(211,230)(212,229)(213,228)(214,227)(215,226)(216,240)(217,239)(218,238)(219,237)(220,236)(221,235)(222,234)(223,233)(224,232)(225,231)>;
G:=Group( (1,219,32,200,20,238,58,189)(2,220,33,201,21,239,59,190)(3,221,34,202,22,240,60,191)(4,222,35,203,23,226,46,192)(5,223,36,204,24,227,47,193)(6,224,37,205,25,228,48,194)(7,225,38,206,26,229,49,195)(8,211,39,207,27,230,50,181)(9,212,40,208,28,231,51,182)(10,213,41,209,29,232,52,183)(11,214,42,210,30,233,53,184)(12,215,43,196,16,234,54,185)(13,216,44,197,17,235,55,186)(14,217,45,198,18,236,56,187)(15,218,31,199,19,237,57,188)(61,154,110,121,80,173,91,143)(62,155,111,122,81,174,92,144)(63,156,112,123,82,175,93,145)(64,157,113,124,83,176,94,146)(65,158,114,125,84,177,95,147)(66,159,115,126,85,178,96,148)(67,160,116,127,86,179,97,149)(68,161,117,128,87,180,98,150)(69,162,118,129,88,166,99,136)(70,163,119,130,89,167,100,137)(71,164,120,131,90,168,101,138)(72,165,106,132,76,169,102,139)(73,151,107,133,77,170,103,140)(74,152,108,134,78,171,104,141)(75,153,109,135,79,172,105,142), (1,77,20,73)(2,78,21,74)(3,79,22,75)(4,80,23,61)(5,81,24,62)(6,82,25,63)(7,83,26,64)(8,84,27,65)(9,85,28,66)(10,86,29,67)(11,87,30,68)(12,88,16,69)(13,89,17,70)(14,90,18,71)(15,76,19,72)(31,106,57,102)(32,107,58,103)(33,108,59,104)(34,109,60,105)(35,110,46,91)(36,111,47,92)(37,112,48,93)(38,113,49,94)(39,114,50,95)(40,115,51,96)(41,116,52,97)(42,117,53,98)(43,118,54,99)(44,119,55,100)(45,120,56,101)(121,226,143,222)(122,227,144,223)(123,228,145,224)(124,229,146,225)(125,230,147,211)(126,231,148,212)(127,232,149,213)(128,233,150,214)(129,234,136,215)(130,235,137,216)(131,236,138,217)(132,237,139,218)(133,238,140,219)(134,239,141,220)(135,240,142,221)(151,189,170,200)(152,190,171,201)(153,191,172,202)(154,192,173,203)(155,193,174,204)(156,194,175,205)(157,195,176,206)(158,181,177,207)(159,182,178,208)(160,183,179,209)(161,184,180,210)(162,185,166,196)(163,186,167,197)(164,187,168,198)(165,188,169,199), (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), (1,19)(2,18)(3,17)(4,16)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,60)(45,59)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,90)(75,89)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,120)(105,119)(121,136)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)(151,169)(152,168)(153,167)(154,166)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(181,207)(182,206)(183,205)(184,204)(185,203)(186,202)(187,201)(188,200)(189,199)(190,198)(191,197)(192,196)(193,210)(194,209)(195,208)(211,230)(212,229)(213,228)(214,227)(215,226)(216,240)(217,239)(218,238)(219,237)(220,236)(221,235)(222,234)(223,233)(224,232)(225,231) );
G=PermutationGroup([[(1,219,32,200,20,238,58,189),(2,220,33,201,21,239,59,190),(3,221,34,202,22,240,60,191),(4,222,35,203,23,226,46,192),(5,223,36,204,24,227,47,193),(6,224,37,205,25,228,48,194),(7,225,38,206,26,229,49,195),(8,211,39,207,27,230,50,181),(9,212,40,208,28,231,51,182),(10,213,41,209,29,232,52,183),(11,214,42,210,30,233,53,184),(12,215,43,196,16,234,54,185),(13,216,44,197,17,235,55,186),(14,217,45,198,18,236,56,187),(15,218,31,199,19,237,57,188),(61,154,110,121,80,173,91,143),(62,155,111,122,81,174,92,144),(63,156,112,123,82,175,93,145),(64,157,113,124,83,176,94,146),(65,158,114,125,84,177,95,147),(66,159,115,126,85,178,96,148),(67,160,116,127,86,179,97,149),(68,161,117,128,87,180,98,150),(69,162,118,129,88,166,99,136),(70,163,119,130,89,167,100,137),(71,164,120,131,90,168,101,138),(72,165,106,132,76,169,102,139),(73,151,107,133,77,170,103,140),(74,152,108,134,78,171,104,141),(75,153,109,135,79,172,105,142)], [(1,77,20,73),(2,78,21,74),(3,79,22,75),(4,80,23,61),(5,81,24,62),(6,82,25,63),(7,83,26,64),(8,84,27,65),(9,85,28,66),(10,86,29,67),(11,87,30,68),(12,88,16,69),(13,89,17,70),(14,90,18,71),(15,76,19,72),(31,106,57,102),(32,107,58,103),(33,108,59,104),(34,109,60,105),(35,110,46,91),(36,111,47,92),(37,112,48,93),(38,113,49,94),(39,114,50,95),(40,115,51,96),(41,116,52,97),(42,117,53,98),(43,118,54,99),(44,119,55,100),(45,120,56,101),(121,226,143,222),(122,227,144,223),(123,228,145,224),(124,229,146,225),(125,230,147,211),(126,231,148,212),(127,232,149,213),(128,233,150,214),(129,234,136,215),(130,235,137,216),(131,236,138,217),(132,237,139,218),(133,238,140,219),(134,239,141,220),(135,240,142,221),(151,189,170,200),(152,190,171,201),(153,191,172,202),(154,192,173,203),(155,193,174,204),(156,194,175,205),(157,195,176,206),(158,181,177,207),(159,182,178,208),(160,183,179,209),(161,184,180,210),(162,185,166,196),(163,186,167,197),(164,187,168,198),(165,188,169,199)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,52),(38,51),(39,50),(40,49),(41,48),(42,47),(43,46),(44,60),(45,59),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77),(73,76),(74,90),(75,89),(91,118),(92,117),(93,116),(94,115),(95,114),(96,113),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,120),(105,119),(121,136),(122,150),(123,149),(124,148),(125,147),(126,146),(127,145),(128,144),(129,143),(130,142),(131,141),(132,140),(133,139),(134,138),(135,137),(151,169),(152,168),(153,167),(154,166),(155,180),(156,179),(157,178),(158,177),(159,176),(160,175),(161,174),(162,173),(163,172),(164,171),(165,170),(181,207),(182,206),(183,205),(184,204),(185,203),(186,202),(187,201),(188,200),(189,199),(190,198),(191,197),(192,196),(193,210),(194,209),(195,208),(211,230),(212,229),(213,228),(214,227),(215,226),(216,240),(217,239),(218,238),(219,237),(220,236),(221,235),(222,234),(223,233),(224,232),(225,231)]])
63 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 | 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 | 4 | 5 | 5 | 6 | 8 | 8 | 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 | 15 | 15 | 2 | 2 | 4 | 4 | 30 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 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 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | Q16 | D10 | D10 | D15 | D30 | D30 | S3×D4 | D4×D5 | S3×Q16 | D5×Q16 | D4×D15 | Q16×D15 |
| kernel | Q16×D15 | C8×D15 | Dic60 | C15⋊7Q16 | C15×Q16 | Q8×D15 | C5×Q16 | Dic15 | D30 | C3×Q16 | C40 | C5×Q8 | D15 | C24 | C3×Q8 | Q16 | C8 | Q8 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of Q16×D15 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 55 |
| 0 | 0 | 92 | 219 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 207 | 219 |
| 0 | 0 | 195 | 34 |
| 30 | 110 | 0 | 0 |
| 195 | 225 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 63 | 157 | 0 | 0 |
| 162 | 178 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,92,0,0,55,219],[1,0,0,0,0,1,0,0,0,0,207,195,0,0,219,34],[30,195,0,0,110,225,0,0,0,0,1,0,0,0,0,1],[63,162,0,0,157,178,0,0,0,0,240,0,0,0,0,240] >;
Q16×D15 in GAP, Magma, Sage, TeX
Q_{16}\times D_{15} % in TeX
G:=Group("Q16xD15"); // GroupNames label
G:=SmallGroup(480,882);
// by ID
G=gap.SmallGroup(480,882);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,135,100,346,185,80,2693,18822]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations