metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊16D4, C15⋊29(C4×D4), C5⋊4(D4×Dic3), C6.87(D4×D5), C10.88(S3×D4), C5⋊D4⋊2Dic3, D10⋊5(C2×Dic3), C30.239(C2×D4), C22⋊3(D5×Dic3), C23.51(S3×D5), Dic5⋊3(C2×Dic3), C6.D4⋊10D5, C3⋊7(Dic5⋊4D4), C30.Q8⋊35C2, (C22×D5).65D6, (C22×C10).53D6, (C22×C6).38D10, (Dic3×Dic5)⋊37C2, C30.153(C4○D4), C2.5(D10⋊D6), D10⋊Dic3⋊32C2, C6.83(D4⋊2D5), (C2×C30).201C23, C30.149(C22×C4), (C2×Dic5).131D6, C10.83(D4⋊2S3), (C2×Dic3).124D10, C2.6(C30.C23), (C22×Dic15)⋊14C2, (C22×C30).63C22, C10.33(C22×Dic3), (C6×Dic5).117C22, (C2×Dic15).227C22, (C10×Dic3).117C22, (C2×C6)⋊3(C4×D5), C6.96(C2×C4×D5), (C3×C5⋊D4)⋊4C4, (C2×C30)⋊20(C2×C4), (C6×D5)⋊10(C2×C4), (C2×D5×Dic3)⋊18C2, (C6×C5⋊D4).7C2, (C2×C5⋊D4).6S3, C2.20(C2×D5×Dic3), C22.89(C2×S3×D5), (C3×Dic5)⋊6(C2×C4), (C2×C10)⋊9(C2×Dic3), (D5×C2×C6).52C22, (C5×C6.D4)⋊12C2, (C2×C6).213(C22×D5), (C2×C10).213(C22×S3), SmallGroup(480,635)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊16D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a19, bc=cb, bd=db, dcd=c-1 >
Subgroups: 828 in 188 conjugacy classes, 70 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic3, C4⋊Dic3, C6.D4, C6.D4, C22×Dic3, C6×D4, C5×Dic3, C3×Dic5, Dic15, Dic15, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, D4×Dic3, D5×Dic3, C6×Dic5, C3×C5⋊D4, C10×Dic3, C2×Dic15, C2×Dic15, D5×C2×C6, C22×C30, Dic5⋊4D4, Dic3×Dic5, D10⋊Dic3, C30.Q8, C5×C6.D4, C2×D5×Dic3, C6×C5⋊D4, C22×Dic15, Dic15⋊16D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, Dic3, D6, C22×C4, C2×D4, C4○D4, D10, C2×Dic3, C22×S3, C4×D4, C4×D5, C22×D5, S3×D4, D4⋊2S3, C22×Dic3, S3×D5, C2×C4×D5, D4×D5, D4⋊2D5, D4×Dic3, D5×Dic3, C2×S3×D5, Dic5⋊4D4, C2×D5×Dic3, C30.C23, D10⋊D6, Dic15⋊16D4
►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 59 16 44)(2 58 17 43)(3 57 18 42)(4 56 19 41)(5 55 20 40)(6 54 21 39)(7 53 22 38)(8 52 23 37)(9 51 24 36)(10 50 25 35)(11 49 26 34)(12 48 27 33)(13 47 28 32)(14 46 29 31)(15 45 30 60)(61 141 76 126)(62 140 77 125)(63 139 78 124)(64 138 79 123)(65 137 80 122)(66 136 81 121)(67 135 82 150)(68 134 83 149)(69 133 84 148)(70 132 85 147)(71 131 86 146)(72 130 87 145)(73 129 88 144)(74 128 89 143)(75 127 90 142)(91 174 106 159)(92 173 107 158)(93 172 108 157)(94 171 109 156)(95 170 110 155)(96 169 111 154)(97 168 112 153)(98 167 113 152)(99 166 114 151)(100 165 115 180)(101 164 116 179)(102 163 117 178)(103 162 118 177)(104 161 119 176)(105 160 120 175)(181 233 196 218)(182 232 197 217)(183 231 198 216)(184 230 199 215)(185 229 200 214)(186 228 201 213)(187 227 202 212)(188 226 203 211)(189 225 204 240)(190 224 205 239)(191 223 206 238)(192 222 207 237)(193 221 208 236)(194 220 209 235)(195 219 210 234)
(1 111 187 67)(2 100 188 86)(3 119 189 75)(4 108 190 64)(5 97 191 83)(6 116 192 72)(7 105 193 61)(8 94 194 80)(9 113 195 69)(10 102 196 88)(11 91 197 77)(12 110 198 66)(13 99 199 85)(14 118 200 74)(15 107 201 63)(16 96 202 82)(17 115 203 71)(18 104 204 90)(19 93 205 79)(20 112 206 68)(21 101 207 87)(22 120 208 76)(23 109 209 65)(24 98 210 84)(25 117 181 73)(26 106 182 62)(27 95 183 81)(28 114 184 70)(29 103 185 89)(30 92 186 78)(31 162 229 143)(32 151 230 132)(33 170 231 121)(34 159 232 140)(35 178 233 129)(36 167 234 148)(37 156 235 137)(38 175 236 126)(39 164 237 145)(40 153 238 134)(41 172 239 123)(42 161 240 142)(43 180 211 131)(44 169 212 150)(45 158 213 139)(46 177 214 128)(47 166 215 147)(48 155 216 136)(49 174 217 125)(50 163 218 144)(51 152 219 133)(52 171 220 122)(53 160 221 141)(54 179 222 130)(55 168 223 149)(56 157 224 138)(57 176 225 127)(58 165 226 146)(59 154 227 135)(60 173 228 124)
(1 187)(2 206)(3 195)(4 184)(5 203)(6 192)(7 181)(8 200)(9 189)(10 208)(11 197)(12 186)(13 205)(14 194)(15 183)(16 202)(17 191)(18 210)(19 199)(20 188)(21 207)(22 196)(23 185)(24 204)(25 193)(26 182)(27 201)(28 190)(29 209)(30 198)(31 235)(32 224)(33 213)(34 232)(35 221)(36 240)(37 229)(38 218)(39 237)(40 226)(41 215)(42 234)(43 223)(44 212)(45 231)(46 220)(47 239)(48 228)(49 217)(50 236)(51 225)(52 214)(53 233)(54 222)(55 211)(56 230)(57 219)(58 238)(59 227)(60 216)(61 73)(63 81)(64 70)(65 89)(66 78)(68 86)(69 75)(71 83)(74 80)(76 88)(79 85)(84 90)(92 110)(93 99)(94 118)(95 107)(97 115)(98 104)(100 112)(102 120)(103 109)(105 117)(108 114)(113 119)(121 139)(122 128)(123 147)(124 136)(126 144)(127 133)(129 141)(131 149)(132 138)(134 146)(137 143)(142 148)(151 157)(152 176)(153 165)(155 173)(156 162)(158 170)(160 178)(161 167)(163 175)(166 172)(168 180)(171 177)
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), (1,59,16,44)(2,58,17,43)(3,57,18,42)(4,56,19,41)(5,55,20,40)(6,54,21,39)(7,53,22,38)(8,52,23,37)(9,51,24,36)(10,50,25,35)(11,49,26,34)(12,48,27,33)(13,47,28,32)(14,46,29,31)(15,45,30,60)(61,141,76,126)(62,140,77,125)(63,139,78,124)(64,138,79,123)(65,137,80,122)(66,136,81,121)(67,135,82,150)(68,134,83,149)(69,133,84,148)(70,132,85,147)(71,131,86,146)(72,130,87,145)(73,129,88,144)(74,128,89,143)(75,127,90,142)(91,174,106,159)(92,173,107,158)(93,172,108,157)(94,171,109,156)(95,170,110,155)(96,169,111,154)(97,168,112,153)(98,167,113,152)(99,166,114,151)(100,165,115,180)(101,164,116,179)(102,163,117,178)(103,162,118,177)(104,161,119,176)(105,160,120,175)(181,233,196,218)(182,232,197,217)(183,231,198,216)(184,230,199,215)(185,229,200,214)(186,228,201,213)(187,227,202,212)(188,226,203,211)(189,225,204,240)(190,224,205,239)(191,223,206,238)(192,222,207,237)(193,221,208,236)(194,220,209,235)(195,219,210,234), (1,111,187,67)(2,100,188,86)(3,119,189,75)(4,108,190,64)(5,97,191,83)(6,116,192,72)(7,105,193,61)(8,94,194,80)(9,113,195,69)(10,102,196,88)(11,91,197,77)(12,110,198,66)(13,99,199,85)(14,118,200,74)(15,107,201,63)(16,96,202,82)(17,115,203,71)(18,104,204,90)(19,93,205,79)(20,112,206,68)(21,101,207,87)(22,120,208,76)(23,109,209,65)(24,98,210,84)(25,117,181,73)(26,106,182,62)(27,95,183,81)(28,114,184,70)(29,103,185,89)(30,92,186,78)(31,162,229,143)(32,151,230,132)(33,170,231,121)(34,159,232,140)(35,178,233,129)(36,167,234,148)(37,156,235,137)(38,175,236,126)(39,164,237,145)(40,153,238,134)(41,172,239,123)(42,161,240,142)(43,180,211,131)(44,169,212,150)(45,158,213,139)(46,177,214,128)(47,166,215,147)(48,155,216,136)(49,174,217,125)(50,163,218,144)(51,152,219,133)(52,171,220,122)(53,160,221,141)(54,179,222,130)(55,168,223,149)(56,157,224,138)(57,176,225,127)(58,165,226,146)(59,154,227,135)(60,173,228,124), (1,187)(2,206)(3,195)(4,184)(5,203)(6,192)(7,181)(8,200)(9,189)(10,208)(11,197)(12,186)(13,205)(14,194)(15,183)(16,202)(17,191)(18,210)(19,199)(20,188)(21,207)(22,196)(23,185)(24,204)(25,193)(26,182)(27,201)(28,190)(29,209)(30,198)(31,235)(32,224)(33,213)(34,232)(35,221)(36,240)(37,229)(38,218)(39,237)(40,226)(41,215)(42,234)(43,223)(44,212)(45,231)(46,220)(47,239)(48,228)(49,217)(50,236)(51,225)(52,214)(53,233)(54,222)(55,211)(56,230)(57,219)(58,238)(59,227)(60,216)(61,73)(63,81)(64,70)(65,89)(66,78)(68,86)(69,75)(71,83)(74,80)(76,88)(79,85)(84,90)(92,110)(93,99)(94,118)(95,107)(97,115)(98,104)(100,112)(102,120)(103,109)(105,117)(108,114)(113,119)(121,139)(122,128)(123,147)(124,136)(126,144)(127,133)(129,141)(131,149)(132,138)(134,146)(137,143)(142,148)(151,157)(152,176)(153,165)(155,173)(156,162)(158,170)(160,178)(161,167)(163,175)(166,172)(168,180)(171,177)>;
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), (1,59,16,44)(2,58,17,43)(3,57,18,42)(4,56,19,41)(5,55,20,40)(6,54,21,39)(7,53,22,38)(8,52,23,37)(9,51,24,36)(10,50,25,35)(11,49,26,34)(12,48,27,33)(13,47,28,32)(14,46,29,31)(15,45,30,60)(61,141,76,126)(62,140,77,125)(63,139,78,124)(64,138,79,123)(65,137,80,122)(66,136,81,121)(67,135,82,150)(68,134,83,149)(69,133,84,148)(70,132,85,147)(71,131,86,146)(72,130,87,145)(73,129,88,144)(74,128,89,143)(75,127,90,142)(91,174,106,159)(92,173,107,158)(93,172,108,157)(94,171,109,156)(95,170,110,155)(96,169,111,154)(97,168,112,153)(98,167,113,152)(99,166,114,151)(100,165,115,180)(101,164,116,179)(102,163,117,178)(103,162,118,177)(104,161,119,176)(105,160,120,175)(181,233,196,218)(182,232,197,217)(183,231,198,216)(184,230,199,215)(185,229,200,214)(186,228,201,213)(187,227,202,212)(188,226,203,211)(189,225,204,240)(190,224,205,239)(191,223,206,238)(192,222,207,237)(193,221,208,236)(194,220,209,235)(195,219,210,234), (1,111,187,67)(2,100,188,86)(3,119,189,75)(4,108,190,64)(5,97,191,83)(6,116,192,72)(7,105,193,61)(8,94,194,80)(9,113,195,69)(10,102,196,88)(11,91,197,77)(12,110,198,66)(13,99,199,85)(14,118,200,74)(15,107,201,63)(16,96,202,82)(17,115,203,71)(18,104,204,90)(19,93,205,79)(20,112,206,68)(21,101,207,87)(22,120,208,76)(23,109,209,65)(24,98,210,84)(25,117,181,73)(26,106,182,62)(27,95,183,81)(28,114,184,70)(29,103,185,89)(30,92,186,78)(31,162,229,143)(32,151,230,132)(33,170,231,121)(34,159,232,140)(35,178,233,129)(36,167,234,148)(37,156,235,137)(38,175,236,126)(39,164,237,145)(40,153,238,134)(41,172,239,123)(42,161,240,142)(43,180,211,131)(44,169,212,150)(45,158,213,139)(46,177,214,128)(47,166,215,147)(48,155,216,136)(49,174,217,125)(50,163,218,144)(51,152,219,133)(52,171,220,122)(53,160,221,141)(54,179,222,130)(55,168,223,149)(56,157,224,138)(57,176,225,127)(58,165,226,146)(59,154,227,135)(60,173,228,124), (1,187)(2,206)(3,195)(4,184)(5,203)(6,192)(7,181)(8,200)(9,189)(10,208)(11,197)(12,186)(13,205)(14,194)(15,183)(16,202)(17,191)(18,210)(19,199)(20,188)(21,207)(22,196)(23,185)(24,204)(25,193)(26,182)(27,201)(28,190)(29,209)(30,198)(31,235)(32,224)(33,213)(34,232)(35,221)(36,240)(37,229)(38,218)(39,237)(40,226)(41,215)(42,234)(43,223)(44,212)(45,231)(46,220)(47,239)(48,228)(49,217)(50,236)(51,225)(52,214)(53,233)(54,222)(55,211)(56,230)(57,219)(58,238)(59,227)(60,216)(61,73)(63,81)(64,70)(65,89)(66,78)(68,86)(69,75)(71,83)(74,80)(76,88)(79,85)(84,90)(92,110)(93,99)(94,118)(95,107)(97,115)(98,104)(100,112)(102,120)(103,109)(105,117)(108,114)(113,119)(121,139)(122,128)(123,147)(124,136)(126,144)(127,133)(129,141)(131,149)(132,138)(134,146)(137,143)(142,148)(151,157)(152,176)(153,165)(155,173)(156,162)(158,170)(160,178)(161,167)(163,175)(166,172)(168,180)(171,177) );
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)], [(1,59,16,44),(2,58,17,43),(3,57,18,42),(4,56,19,41),(5,55,20,40),(6,54,21,39),(7,53,22,38),(8,52,23,37),(9,51,24,36),(10,50,25,35),(11,49,26,34),(12,48,27,33),(13,47,28,32),(14,46,29,31),(15,45,30,60),(61,141,76,126),(62,140,77,125),(63,139,78,124),(64,138,79,123),(65,137,80,122),(66,136,81,121),(67,135,82,150),(68,134,83,149),(69,133,84,148),(70,132,85,147),(71,131,86,146),(72,130,87,145),(73,129,88,144),(74,128,89,143),(75,127,90,142),(91,174,106,159),(92,173,107,158),(93,172,108,157),(94,171,109,156),(95,170,110,155),(96,169,111,154),(97,168,112,153),(98,167,113,152),(99,166,114,151),(100,165,115,180),(101,164,116,179),(102,163,117,178),(103,162,118,177),(104,161,119,176),(105,160,120,175),(181,233,196,218),(182,232,197,217),(183,231,198,216),(184,230,199,215),(185,229,200,214),(186,228,201,213),(187,227,202,212),(188,226,203,211),(189,225,204,240),(190,224,205,239),(191,223,206,238),(192,222,207,237),(193,221,208,236),(194,220,209,235),(195,219,210,234)], [(1,111,187,67),(2,100,188,86),(3,119,189,75),(4,108,190,64),(5,97,191,83),(6,116,192,72),(7,105,193,61),(8,94,194,80),(9,113,195,69),(10,102,196,88),(11,91,197,77),(12,110,198,66),(13,99,199,85),(14,118,200,74),(15,107,201,63),(16,96,202,82),(17,115,203,71),(18,104,204,90),(19,93,205,79),(20,112,206,68),(21,101,207,87),(22,120,208,76),(23,109,209,65),(24,98,210,84),(25,117,181,73),(26,106,182,62),(27,95,183,81),(28,114,184,70),(29,103,185,89),(30,92,186,78),(31,162,229,143),(32,151,230,132),(33,170,231,121),(34,159,232,140),(35,178,233,129),(36,167,234,148),(37,156,235,137),(38,175,236,126),(39,164,237,145),(40,153,238,134),(41,172,239,123),(42,161,240,142),(43,180,211,131),(44,169,212,150),(45,158,213,139),(46,177,214,128),(47,166,215,147),(48,155,216,136),(49,174,217,125),(50,163,218,144),(51,152,219,133),(52,171,220,122),(53,160,221,141),(54,179,222,130),(55,168,223,149),(56,157,224,138),(57,176,225,127),(58,165,226,146),(59,154,227,135),(60,173,228,124)], [(1,187),(2,206),(3,195),(4,184),(5,203),(6,192),(7,181),(8,200),(9,189),(10,208),(11,197),(12,186),(13,205),(14,194),(15,183),(16,202),(17,191),(18,210),(19,199),(20,188),(21,207),(22,196),(23,185),(24,204),(25,193),(26,182),(27,201),(28,190),(29,209),(30,198),(31,235),(32,224),(33,213),(34,232),(35,221),(36,240),(37,229),(38,218),(39,237),(40,226),(41,215),(42,234),(43,223),(44,212),(45,231),(46,220),(47,239),(48,228),(49,217),(50,236),(51,225),(52,214),(53,233),(54,222),(55,211),(56,230),(57,219),(58,238),(59,227),(60,216),(61,73),(63,81),(64,70),(65,89),(66,78),(68,86),(69,75),(71,83),(74,80),(76,88),(79,85),(84,90),(92,110),(93,99),(94,118),(95,107),(97,115),(98,104),(100,112),(102,120),(103,109),(105,117),(108,114),(113,119),(121,139),(122,128),(123,147),(124,136),(126,144),(127,133),(129,141),(131,149),(132,138),(134,146),(137,143),(142,148),(151,157),(152,176),(153,165),(155,173),(156,162),(158,170),(160,178),(161,167),(163,175),(166,172),(168,180),(171,177)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | + | - | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | Dic3 | D6 | D6 | C4○D4 | D10 | D10 | C4×D5 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | D5×Dic3 | C2×S3×D5 | C30.C23 | D10⋊D6 |
| kernel | Dic15⋊16D4 | Dic3×Dic5 | D10⋊Dic3 | C30.Q8 | C5×C6.D4 | C2×D5×Dic3 | C6×C5⋊D4 | C22×Dic15 | C3×C5⋊D4 | C2×C5⋊D4 | Dic15 | C6.D4 | C2×Dic5 | C5⋊D4 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C2×C6 | C10 | C10 | C23 | C6 | C6 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic15⋊16D4 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 46 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 51 |
| 0 | 0 | 0 | 0 | 26 | 25 |
| 0 | 30 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,60,18,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,46,0,0,0,0,0,50,0,0,0,0,0,0,36,26,0,0,0,0,51,25],[0,2,0,0,0,0,30,0,0,0,0,0,0,0,60,18,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,43,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic15⋊16D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_{16}D_4 % in TeX
G:=Group("Dic15:16D4"); // GroupNames label
G:=SmallGroup(480,635);
// by ID
G=gap.SmallGroup(480,635);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations