metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊19D4, C23.18D30, C15⋊34(C4×D4), C15⋊7D4⋊3C4, C6.97(D4×D5), C2.2(D4×D15), D30⋊19(C2×C4), C22⋊C4⋊7D15, C10.99(S3×D4), (C2×C4).27D30, C22⋊2(C4×D15), (C2×C20).207D6, C30.305(C2×D4), C30.4Q8⋊9C2, D30⋊3C4⋊10C2, (C4×Dic15)⋊17C2, Dic15⋊13(C2×C4), (C2×C12).204D10, C5⋊5(Dic3⋊4D4), C3⋊4(Dic5⋊4D4), (C22×C10).71D6, (C22×C6).56D10, C30.216(C4○D4), C2.2(D4⋊2D15), C6.91(D4⋊2D5), (C2×C30).278C23, C30.158(C22×C4), (C2×C60).172C22, (C22×Dic15)⋊1C2, C10.91(D4⋊2S3), (C22×C30).12C22, C22.14(C22×D15), (C22×D15).79C22, (C2×Dic15).238C22, (C2×C6)⋊6(C4×D5), C2.9(C2×C4×D15), C6.63(C2×C4×D5), (C2×C4×D15)⋊15C2, C10.95(S3×C2×C4), (C2×C30)⋊12(C2×C4), (C2×C10)⋊15(C4×S3), (C5×C22⋊C4)⋊9S3, (C3×C22⋊C4)⋊9D5, (C2×C15⋊7D4).2C2, (C15×C22⋊C4)⋊11C2, (C2×C6).274(C22×D5), (C2×C10).273(C22×S3), SmallGroup(480,846)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊19D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1028 in 188 conjugacy classes, 65 normal (47 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C4×D4, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, Dic15, Dic15, C60, D30, D30, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, Dic3⋊4D4, C4×D15, C2×Dic15, C2×Dic15, C15⋊7D4, C2×C60, C22×D15, C22×C30, Dic5⋊4D4, C4×Dic15, C30.4Q8, D30⋊3C4, C15×C22⋊C4, C2×C4×D15, C22×Dic15, C2×C15⋊7D4, Dic15⋊19D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C22×S3, D15, C4×D4, C4×D5, C22×D5, S3×C2×C4, S3×D4, D4⋊2S3, D30, C2×C4×D5, D4×D5, D4⋊2D5, Dic3⋊4D4, C4×D15, C22×D15, Dic5⋊4D4, C2×C4×D15, D4×D15, D4⋊2D15, Dic15⋊19D4
►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)
(1 133 16 148)(2 132 17 147)(3 131 18 146)(4 130 19 145)(5 129 20 144)(6 128 21 143)(7 127 22 142)(8 126 23 141)(9 125 24 140)(10 124 25 139)(11 123 26 138)(12 122 27 137)(13 121 28 136)(14 150 29 135)(15 149 30 134)(31 183 46 198)(32 182 47 197)(33 181 48 196)(34 210 49 195)(35 209 50 194)(36 208 51 193)(37 207 52 192)(38 206 53 191)(39 205 54 190)(40 204 55 189)(41 203 56 188)(42 202 57 187)(43 201 58 186)(44 200 59 185)(45 199 60 184)(61 212 76 227)(62 211 77 226)(63 240 78 225)(64 239 79 224)(65 238 80 223)(66 237 81 222)(67 236 82 221)(68 235 83 220)(69 234 84 219)(70 233 85 218)(71 232 86 217)(72 231 87 216)(73 230 88 215)(74 229 89 214)(75 228 90 213)(91 152 106 167)(92 151 107 166)(93 180 108 165)(94 179 109 164)(95 178 110 163)(96 177 111 162)(97 176 112 161)(98 175 113 160)(99 174 114 159)(100 173 115 158)(101 172 116 157)(102 171 117 156)(103 170 118 155)(104 169 119 154)(105 168 120 153)
(1 39 115 86)(2 38 116 85)(3 37 117 84)(4 36 118 83)(5 35 119 82)(6 34 120 81)(7 33 91 80)(8 32 92 79)(9 31 93 78)(10 60 94 77)(11 59 95 76)(12 58 96 75)(13 57 97 74)(14 56 98 73)(15 55 99 72)(16 54 100 71)(17 53 101 70)(18 52 102 69)(19 51 103 68)(20 50 104 67)(21 49 105 66)(22 48 106 65)(23 47 107 64)(24 46 108 63)(25 45 109 62)(26 44 110 61)(27 43 111 90)(28 42 112 89)(29 41 113 88)(30 40 114 87)(121 187 176 229)(122 186 177 228)(123 185 178 227)(124 184 179 226)(125 183 180 225)(126 182 151 224)(127 181 152 223)(128 210 153 222)(129 209 154 221)(130 208 155 220)(131 207 156 219)(132 206 157 218)(133 205 158 217)(134 204 159 216)(135 203 160 215)(136 202 161 214)(137 201 162 213)(138 200 163 212)(139 199 164 211)(140 198 165 240)(141 197 166 239)(142 196 167 238)(143 195 168 237)(144 194 169 236)(145 193 170 235)(146 192 171 234)(147 191 172 233)(148 190 173 232)(149 189 174 231)(150 188 175 230)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 105)(60 106)(121 235)(122 236)(123 237)(124 238)(125 239)(126 240)(127 211)(128 212)(129 213)(130 214)(131 215)(132 216)(133 217)(134 218)(135 219)(136 220)(137 221)(138 222)(139 223)(140 224)(141 225)(142 226)(143 227)(144 228)(145 229)(146 230)(147 231)(148 232)(149 233)(150 234)(151 198)(152 199)(153 200)(154 201)(155 202)(156 203)(157 204)(158 205)(159 206)(160 207)(161 208)(162 209)(163 210)(164 181)(165 182)(166 183)(167 184)(168 185)(169 186)(170 187)(171 188)(172 189)(173 190)(174 191)(175 192)(176 193)(177 194)(178 195)(179 196)(180 197)
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), (1,133,16,148)(2,132,17,147)(3,131,18,146)(4,130,19,145)(5,129,20,144)(6,128,21,143)(7,127,22,142)(8,126,23,141)(9,125,24,140)(10,124,25,139)(11,123,26,138)(12,122,27,137)(13,121,28,136)(14,150,29,135)(15,149,30,134)(31,183,46,198)(32,182,47,197)(33,181,48,196)(34,210,49,195)(35,209,50,194)(36,208,51,193)(37,207,52,192)(38,206,53,191)(39,205,54,190)(40,204,55,189)(41,203,56,188)(42,202,57,187)(43,201,58,186)(44,200,59,185)(45,199,60,184)(61,212,76,227)(62,211,77,226)(63,240,78,225)(64,239,79,224)(65,238,80,223)(66,237,81,222)(67,236,82,221)(68,235,83,220)(69,234,84,219)(70,233,85,218)(71,232,86,217)(72,231,87,216)(73,230,88,215)(74,229,89,214)(75,228,90,213)(91,152,106,167)(92,151,107,166)(93,180,108,165)(94,179,109,164)(95,178,110,163)(96,177,111,162)(97,176,112,161)(98,175,113,160)(99,174,114,159)(100,173,115,158)(101,172,116,157)(102,171,117,156)(103,170,118,155)(104,169,119,154)(105,168,120,153), (1,39,115,86)(2,38,116,85)(3,37,117,84)(4,36,118,83)(5,35,119,82)(6,34,120,81)(7,33,91,80)(8,32,92,79)(9,31,93,78)(10,60,94,77)(11,59,95,76)(12,58,96,75)(13,57,97,74)(14,56,98,73)(15,55,99,72)(16,54,100,71)(17,53,101,70)(18,52,102,69)(19,51,103,68)(20,50,104,67)(21,49,105,66)(22,48,106,65)(23,47,107,64)(24,46,108,63)(25,45,109,62)(26,44,110,61)(27,43,111,90)(28,42,112,89)(29,41,113,88)(30,40,114,87)(121,187,176,229)(122,186,177,228)(123,185,178,227)(124,184,179,226)(125,183,180,225)(126,182,151,224)(127,181,152,223)(128,210,153,222)(129,209,154,221)(130,208,155,220)(131,207,156,219)(132,206,157,218)(133,205,158,217)(134,204,159,216)(135,203,160,215)(136,202,161,214)(137,201,162,213)(138,200,163,212)(139,199,164,211)(140,198,165,240)(141,197,166,239)(142,196,167,238)(143,195,168,237)(144,194,169,236)(145,193,170,235)(146,192,171,234)(147,191,172,233)(148,190,173,232)(149,189,174,231)(150,188,175,230), (1,86)(2,87)(3,88)(4,89)(5,90)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,105)(60,106)(121,235)(122,236)(123,237)(124,238)(125,239)(126,240)(127,211)(128,212)(129,213)(130,214)(131,215)(132,216)(133,217)(134,218)(135,219)(136,220)(137,221)(138,222)(139,223)(140,224)(141,225)(142,226)(143,227)(144,228)(145,229)(146,230)(147,231)(148,232)(149,233)(150,234)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)(163,210)(164,181)(165,182)(166,183)(167,184)(168,185)(169,186)(170,187)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)>;
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), (1,133,16,148)(2,132,17,147)(3,131,18,146)(4,130,19,145)(5,129,20,144)(6,128,21,143)(7,127,22,142)(8,126,23,141)(9,125,24,140)(10,124,25,139)(11,123,26,138)(12,122,27,137)(13,121,28,136)(14,150,29,135)(15,149,30,134)(31,183,46,198)(32,182,47,197)(33,181,48,196)(34,210,49,195)(35,209,50,194)(36,208,51,193)(37,207,52,192)(38,206,53,191)(39,205,54,190)(40,204,55,189)(41,203,56,188)(42,202,57,187)(43,201,58,186)(44,200,59,185)(45,199,60,184)(61,212,76,227)(62,211,77,226)(63,240,78,225)(64,239,79,224)(65,238,80,223)(66,237,81,222)(67,236,82,221)(68,235,83,220)(69,234,84,219)(70,233,85,218)(71,232,86,217)(72,231,87,216)(73,230,88,215)(74,229,89,214)(75,228,90,213)(91,152,106,167)(92,151,107,166)(93,180,108,165)(94,179,109,164)(95,178,110,163)(96,177,111,162)(97,176,112,161)(98,175,113,160)(99,174,114,159)(100,173,115,158)(101,172,116,157)(102,171,117,156)(103,170,118,155)(104,169,119,154)(105,168,120,153), (1,39,115,86)(2,38,116,85)(3,37,117,84)(4,36,118,83)(5,35,119,82)(6,34,120,81)(7,33,91,80)(8,32,92,79)(9,31,93,78)(10,60,94,77)(11,59,95,76)(12,58,96,75)(13,57,97,74)(14,56,98,73)(15,55,99,72)(16,54,100,71)(17,53,101,70)(18,52,102,69)(19,51,103,68)(20,50,104,67)(21,49,105,66)(22,48,106,65)(23,47,107,64)(24,46,108,63)(25,45,109,62)(26,44,110,61)(27,43,111,90)(28,42,112,89)(29,41,113,88)(30,40,114,87)(121,187,176,229)(122,186,177,228)(123,185,178,227)(124,184,179,226)(125,183,180,225)(126,182,151,224)(127,181,152,223)(128,210,153,222)(129,209,154,221)(130,208,155,220)(131,207,156,219)(132,206,157,218)(133,205,158,217)(134,204,159,216)(135,203,160,215)(136,202,161,214)(137,201,162,213)(138,200,163,212)(139,199,164,211)(140,198,165,240)(141,197,166,239)(142,196,167,238)(143,195,168,237)(144,194,169,236)(145,193,170,235)(146,192,171,234)(147,191,172,233)(148,190,173,232)(149,189,174,231)(150,188,175,230), (1,86)(2,87)(3,88)(4,89)(5,90)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,105)(60,106)(121,235)(122,236)(123,237)(124,238)(125,239)(126,240)(127,211)(128,212)(129,213)(130,214)(131,215)(132,216)(133,217)(134,218)(135,219)(136,220)(137,221)(138,222)(139,223)(140,224)(141,225)(142,226)(143,227)(144,228)(145,229)(146,230)(147,231)(148,232)(149,233)(150,234)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)(163,210)(164,181)(165,182)(166,183)(167,184)(168,185)(169,186)(170,187)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197) );
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)], [(1,133,16,148),(2,132,17,147),(3,131,18,146),(4,130,19,145),(5,129,20,144),(6,128,21,143),(7,127,22,142),(8,126,23,141),(9,125,24,140),(10,124,25,139),(11,123,26,138),(12,122,27,137),(13,121,28,136),(14,150,29,135),(15,149,30,134),(31,183,46,198),(32,182,47,197),(33,181,48,196),(34,210,49,195),(35,209,50,194),(36,208,51,193),(37,207,52,192),(38,206,53,191),(39,205,54,190),(40,204,55,189),(41,203,56,188),(42,202,57,187),(43,201,58,186),(44,200,59,185),(45,199,60,184),(61,212,76,227),(62,211,77,226),(63,240,78,225),(64,239,79,224),(65,238,80,223),(66,237,81,222),(67,236,82,221),(68,235,83,220),(69,234,84,219),(70,233,85,218),(71,232,86,217),(72,231,87,216),(73,230,88,215),(74,229,89,214),(75,228,90,213),(91,152,106,167),(92,151,107,166),(93,180,108,165),(94,179,109,164),(95,178,110,163),(96,177,111,162),(97,176,112,161),(98,175,113,160),(99,174,114,159),(100,173,115,158),(101,172,116,157),(102,171,117,156),(103,170,118,155),(104,169,119,154),(105,168,120,153)], [(1,39,115,86),(2,38,116,85),(3,37,117,84),(4,36,118,83),(5,35,119,82),(6,34,120,81),(7,33,91,80),(8,32,92,79),(9,31,93,78),(10,60,94,77),(11,59,95,76),(12,58,96,75),(13,57,97,74),(14,56,98,73),(15,55,99,72),(16,54,100,71),(17,53,101,70),(18,52,102,69),(19,51,103,68),(20,50,104,67),(21,49,105,66),(22,48,106,65),(23,47,107,64),(24,46,108,63),(25,45,109,62),(26,44,110,61),(27,43,111,90),(28,42,112,89),(29,41,113,88),(30,40,114,87),(121,187,176,229),(122,186,177,228),(123,185,178,227),(124,184,179,226),(125,183,180,225),(126,182,151,224),(127,181,152,223),(128,210,153,222),(129,209,154,221),(130,208,155,220),(131,207,156,219),(132,206,157,218),(133,205,158,217),(134,204,159,216),(135,203,160,215),(136,202,161,214),(137,201,162,213),(138,200,163,212),(139,199,164,211),(140,198,165,240),(141,197,166,239),(142,196,167,238),(143,195,168,237),(144,194,169,236),(145,193,170,235),(146,192,171,234),(147,191,172,233),(148,190,173,232),(149,189,174,231),(150,188,175,230)], [(1,86),(2,87),(3,88),(4,89),(5,90),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,107),(32,108),(33,109),(34,110),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(45,91),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(53,99),(54,100),(55,101),(56,102),(57,103),(58,104),(59,105),(60,106),(121,235),(122,236),(123,237),(124,238),(125,239),(126,240),(127,211),(128,212),(129,213),(130,214),(131,215),(132,216),(133,217),(134,218),(135,219),(136,220),(137,221),(138,222),(139,223),(140,224),(141,225),(142,226),(143,227),(144,228),(145,229),(146,230),(147,231),(148,232),(149,233),(150,234),(151,198),(152,199),(153,200),(154,201),(155,202),(156,203),(157,204),(158,205),(159,206),(160,207),(161,208),(162,209),(163,210),(164,181),(165,182),(166,183),(167,184),(168,185),(169,186),(170,187),(171,188),(172,189),(173,190),(174,191),(175,192),(176,193),(177,194),(178,195),(179,196),(180,197)]])
90 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 | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | D15 | C4×D5 | D30 | D30 | C4×D15 | S3×D4 | D4⋊2S3 | D4×D5 | D4⋊2D5 | D4×D15 | D4⋊2D15 |
| kernel | Dic15⋊19D4 | C4×Dic15 | C30.4Q8 | D30⋊3C4 | C15×C22⋊C4 | C2×C4×D15 | C22×Dic15 | C2×C15⋊7D4 | C15⋊7D4 | C5×C22⋊C4 | Dic15 | C3×C22⋊C4 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C2×C10 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic15⋊19D4 ►in GL6(𝔽61)
| 17 | 60 | 0 | 0 | 0 | 0 |
| 45 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 24 | 0 | 0 |
| 0 | 0 | 37 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 57 | 46 | 0 | 0 | 0 | 0 |
| 54 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 44 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 17 | 18 | 0 | 0 | 0 | 0 |
| 45 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 44 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 12 |
| 0 | 0 | 0 | 0 | 10 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [17,45,0,0,0,0,60,1,0,0,0,0,0,0,33,37,0,0,0,0,24,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,54,0,0,0,0,46,4,0,0,0,0,0,0,60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[17,45,0,0,0,0,18,44,0,0,0,0,0,0,60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,10,0,0,0,0,12,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,12,1] >;
Dic15⋊19D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_{19}D_4 % in TeX
G:=Group("Dic15:19D4"); // GroupNames label
G:=SmallGroup(480,846);
// by ID
G=gap.SmallGroup(480,846);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,219,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations