metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊13D4, C15⋊6(C4×D4), D10⋊4(C4×S3), C3⋊D20⋊4C4, C6.56(D4×D5), D30⋊22(C2×C4), Dic3⋊1(C4×D5), C10.58(S3×D4), C3⋊2(D20⋊8C4), Dic3⋊C4⋊13D5, (C2×C20).191D6, C30.125(C2×D4), D10⋊C4⋊10S3, D30⋊4C4⋊10C2, C30.52(C4○D4), (C2×C12).190D10, C5⋊3(Dic3⋊4D4), (C2×C30).86C23, C30.48(C22×C4), (C22×D5).44D6, (Dic3×Dic5)⋊13C2, C2.4(D20⋊S3), C2.1(D10⋊D6), (C2×C60).164C22, C6.31(Q8⋊2D5), (C2×Dic3).91D10, (C2×Dic5).100D6, C10.11(D4⋊2S3), (C6×Dic5).51C22, (C10×Dic3).50C22, (C2×Dic15).200C22, (C22×D15).101C22, (C2×C4×D15)⋊9C2, C6.16(C2×C4×D5), C2.19(C4×S3×D5), (C6×D5)⋊2(C2×C4), C10.48(S3×C2×C4), (C2×D5×Dic3)⋊2C2, C22.41(C2×S3×D5), (C5×Dic3)⋊8(C2×C4), (C2×C4).177(S3×D5), (C2×C3⋊D20).6C2, (D5×C2×C6).13C22, (C5×Dic3⋊C4)⋊13C2, (C3×D10⋊C4)⋊10C2, (C2×C6).98(C22×D5), (C2×C10).98(C22×S3), SmallGroup(480,472)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊13D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=a11, dad=a19, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1036 in 188 conjugacy classes, 60 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, D30, D30, C2×C30, C4×Dic5, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Dic3⋊4D4, D5×Dic3, C3⋊D20, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D20⋊8C4, Dic3×Dic5, D30⋊4C4, C3×D10⋊C4, C5×Dic3⋊C4, C2×D5×Dic3, C2×C3⋊D20, C2×C4×D15, Dic15⋊13D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C22×S3, C4×D4, C4×D5, C22×D5, S3×C2×C4, S3×D4, D4⋊2S3, S3×D5, C2×C4×D5, D4×D5, Q8⋊2D5, Dic3⋊4D4, C2×S3×D5, D20⋊8C4, D20⋊S3, C4×S3×D5, D10⋊D6, Dic15⋊13D4
►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)
(1 180 16 165)(2 179 17 164)(3 178 18 163)(4 177 19 162)(5 176 20 161)(6 175 21 160)(7 174 22 159)(8 173 23 158)(9 172 24 157)(10 171 25 156)(11 170 26 155)(12 169 27 154)(13 168 28 153)(14 167 29 152)(15 166 30 151)(31 141 46 126)(32 140 47 125)(33 139 48 124)(34 138 49 123)(35 137 50 122)(36 136 51 121)(37 135 52 150)(38 134 53 149)(39 133 54 148)(40 132 55 147)(41 131 56 146)(42 130 57 145)(43 129 58 144)(44 128 59 143)(45 127 60 142)(61 194 76 209)(62 193 77 208)(63 192 78 207)(64 191 79 206)(65 190 80 205)(66 189 81 204)(67 188 82 203)(68 187 83 202)(69 186 84 201)(70 185 85 200)(71 184 86 199)(72 183 87 198)(73 182 88 197)(74 181 89 196)(75 210 90 195)(91 215 106 230)(92 214 107 229)(93 213 108 228)(94 212 109 227)(95 211 110 226)(96 240 111 225)(97 239 112 224)(98 238 113 223)(99 237 114 222)(100 236 115 221)(101 235 116 220)(102 234 117 219)(103 233 118 218)(104 232 119 217)(105 231 120 216)
(1 232 45 188)(2 213 46 199)(3 224 47 210)(4 235 48 191)(5 216 49 202)(6 227 50 183)(7 238 51 194)(8 219 52 205)(9 230 53 186)(10 211 54 197)(11 222 55 208)(12 233 56 189)(13 214 57 200)(14 225 58 181)(15 236 59 192)(16 217 60 203)(17 228 31 184)(18 239 32 195)(19 220 33 206)(20 231 34 187)(21 212 35 198)(22 223 36 209)(23 234 37 190)(24 215 38 201)(25 226 39 182)(26 237 40 193)(27 218 41 204)(28 229 42 185)(29 240 43 196)(30 221 44 207)(61 159 98 136)(62 170 99 147)(63 151 100 128)(64 162 101 139)(65 173 102 150)(66 154 103 131)(67 165 104 142)(68 176 105 123)(69 157 106 134)(70 168 107 145)(71 179 108 126)(72 160 109 137)(73 171 110 148)(74 152 111 129)(75 163 112 140)(76 174 113 121)(77 155 114 132)(78 166 115 143)(79 177 116 124)(80 158 117 135)(81 169 118 146)(82 180 119 127)(83 161 120 138)(84 172 91 149)(85 153 92 130)(86 164 93 141)(87 175 94 122)(88 156 95 133)(89 167 96 144)(90 178 97 125)
(1 45)(2 34)(3 53)(4 42)(5 31)(6 50)(7 39)(8 58)(9 47)(10 36)(11 55)(12 44)(13 33)(14 52)(15 41)(16 60)(17 49)(18 38)(19 57)(20 46)(21 35)(22 54)(23 43)(24 32)(25 51)(26 40)(27 59)(28 48)(29 37)(30 56)(61 73)(63 81)(64 70)(65 89)(66 78)(68 86)(69 75)(71 83)(74 80)(76 88)(79 85)(84 90)(91 97)(92 116)(93 105)(95 113)(96 102)(98 110)(100 118)(101 107)(103 115)(106 112)(108 120)(111 117)(121 156)(122 175)(123 164)(124 153)(125 172)(126 161)(127 180)(128 169)(129 158)(130 177)(131 166)(132 155)(133 174)(134 163)(135 152)(136 171)(137 160)(138 179)(139 168)(140 157)(141 176)(142 165)(143 154)(144 173)(145 162)(146 151)(147 170)(148 159)(149 178)(150 167)(181 205)(182 194)(184 202)(185 191)(186 210)(187 199)(189 207)(190 196)(192 204)(195 201)(197 209)(200 206)(211 223)(213 231)(214 220)(215 239)(216 228)(218 236)(219 225)(221 233)(224 230)(226 238)(229 235)(234 240)
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), (1,180,16,165)(2,179,17,164)(3,178,18,163)(4,177,19,162)(5,176,20,161)(6,175,21,160)(7,174,22,159)(8,173,23,158)(9,172,24,157)(10,171,25,156)(11,170,26,155)(12,169,27,154)(13,168,28,153)(14,167,29,152)(15,166,30,151)(31,141,46,126)(32,140,47,125)(33,139,48,124)(34,138,49,123)(35,137,50,122)(36,136,51,121)(37,135,52,150)(38,134,53,149)(39,133,54,148)(40,132,55,147)(41,131,56,146)(42,130,57,145)(43,129,58,144)(44,128,59,143)(45,127,60,142)(61,194,76,209)(62,193,77,208)(63,192,78,207)(64,191,79,206)(65,190,80,205)(66,189,81,204)(67,188,82,203)(68,187,83,202)(69,186,84,201)(70,185,85,200)(71,184,86,199)(72,183,87,198)(73,182,88,197)(74,181,89,196)(75,210,90,195)(91,215,106,230)(92,214,107,229)(93,213,108,228)(94,212,109,227)(95,211,110,226)(96,240,111,225)(97,239,112,224)(98,238,113,223)(99,237,114,222)(100,236,115,221)(101,235,116,220)(102,234,117,219)(103,233,118,218)(104,232,119,217)(105,231,120,216), (1,232,45,188)(2,213,46,199)(3,224,47,210)(4,235,48,191)(5,216,49,202)(6,227,50,183)(7,238,51,194)(8,219,52,205)(9,230,53,186)(10,211,54,197)(11,222,55,208)(12,233,56,189)(13,214,57,200)(14,225,58,181)(15,236,59,192)(16,217,60,203)(17,228,31,184)(18,239,32,195)(19,220,33,206)(20,231,34,187)(21,212,35,198)(22,223,36,209)(23,234,37,190)(24,215,38,201)(25,226,39,182)(26,237,40,193)(27,218,41,204)(28,229,42,185)(29,240,43,196)(30,221,44,207)(61,159,98,136)(62,170,99,147)(63,151,100,128)(64,162,101,139)(65,173,102,150)(66,154,103,131)(67,165,104,142)(68,176,105,123)(69,157,106,134)(70,168,107,145)(71,179,108,126)(72,160,109,137)(73,171,110,148)(74,152,111,129)(75,163,112,140)(76,174,113,121)(77,155,114,132)(78,166,115,143)(79,177,116,124)(80,158,117,135)(81,169,118,146)(82,180,119,127)(83,161,120,138)(84,172,91,149)(85,153,92,130)(86,164,93,141)(87,175,94,122)(88,156,95,133)(89,167,96,144)(90,178,97,125), (1,45)(2,34)(3,53)(4,42)(5,31)(6,50)(7,39)(8,58)(9,47)(10,36)(11,55)(12,44)(13,33)(14,52)(15,41)(16,60)(17,49)(18,38)(19,57)(20,46)(21,35)(22,54)(23,43)(24,32)(25,51)(26,40)(27,59)(28,48)(29,37)(30,56)(61,73)(63,81)(64,70)(65,89)(66,78)(68,86)(69,75)(71,83)(74,80)(76,88)(79,85)(84,90)(91,97)(92,116)(93,105)(95,113)(96,102)(98,110)(100,118)(101,107)(103,115)(106,112)(108,120)(111,117)(121,156)(122,175)(123,164)(124,153)(125,172)(126,161)(127,180)(128,169)(129,158)(130,177)(131,166)(132,155)(133,174)(134,163)(135,152)(136,171)(137,160)(138,179)(139,168)(140,157)(141,176)(142,165)(143,154)(144,173)(145,162)(146,151)(147,170)(148,159)(149,178)(150,167)(181,205)(182,194)(184,202)(185,191)(186,210)(187,199)(189,207)(190,196)(192,204)(195,201)(197,209)(200,206)(211,223)(213,231)(214,220)(215,239)(216,228)(218,236)(219,225)(221,233)(224,230)(226,238)(229,235)(234,240)>;
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), (1,180,16,165)(2,179,17,164)(3,178,18,163)(4,177,19,162)(5,176,20,161)(6,175,21,160)(7,174,22,159)(8,173,23,158)(9,172,24,157)(10,171,25,156)(11,170,26,155)(12,169,27,154)(13,168,28,153)(14,167,29,152)(15,166,30,151)(31,141,46,126)(32,140,47,125)(33,139,48,124)(34,138,49,123)(35,137,50,122)(36,136,51,121)(37,135,52,150)(38,134,53,149)(39,133,54,148)(40,132,55,147)(41,131,56,146)(42,130,57,145)(43,129,58,144)(44,128,59,143)(45,127,60,142)(61,194,76,209)(62,193,77,208)(63,192,78,207)(64,191,79,206)(65,190,80,205)(66,189,81,204)(67,188,82,203)(68,187,83,202)(69,186,84,201)(70,185,85,200)(71,184,86,199)(72,183,87,198)(73,182,88,197)(74,181,89,196)(75,210,90,195)(91,215,106,230)(92,214,107,229)(93,213,108,228)(94,212,109,227)(95,211,110,226)(96,240,111,225)(97,239,112,224)(98,238,113,223)(99,237,114,222)(100,236,115,221)(101,235,116,220)(102,234,117,219)(103,233,118,218)(104,232,119,217)(105,231,120,216), (1,232,45,188)(2,213,46,199)(3,224,47,210)(4,235,48,191)(5,216,49,202)(6,227,50,183)(7,238,51,194)(8,219,52,205)(9,230,53,186)(10,211,54,197)(11,222,55,208)(12,233,56,189)(13,214,57,200)(14,225,58,181)(15,236,59,192)(16,217,60,203)(17,228,31,184)(18,239,32,195)(19,220,33,206)(20,231,34,187)(21,212,35,198)(22,223,36,209)(23,234,37,190)(24,215,38,201)(25,226,39,182)(26,237,40,193)(27,218,41,204)(28,229,42,185)(29,240,43,196)(30,221,44,207)(61,159,98,136)(62,170,99,147)(63,151,100,128)(64,162,101,139)(65,173,102,150)(66,154,103,131)(67,165,104,142)(68,176,105,123)(69,157,106,134)(70,168,107,145)(71,179,108,126)(72,160,109,137)(73,171,110,148)(74,152,111,129)(75,163,112,140)(76,174,113,121)(77,155,114,132)(78,166,115,143)(79,177,116,124)(80,158,117,135)(81,169,118,146)(82,180,119,127)(83,161,120,138)(84,172,91,149)(85,153,92,130)(86,164,93,141)(87,175,94,122)(88,156,95,133)(89,167,96,144)(90,178,97,125), (1,45)(2,34)(3,53)(4,42)(5,31)(6,50)(7,39)(8,58)(9,47)(10,36)(11,55)(12,44)(13,33)(14,52)(15,41)(16,60)(17,49)(18,38)(19,57)(20,46)(21,35)(22,54)(23,43)(24,32)(25,51)(26,40)(27,59)(28,48)(29,37)(30,56)(61,73)(63,81)(64,70)(65,89)(66,78)(68,86)(69,75)(71,83)(74,80)(76,88)(79,85)(84,90)(91,97)(92,116)(93,105)(95,113)(96,102)(98,110)(100,118)(101,107)(103,115)(106,112)(108,120)(111,117)(121,156)(122,175)(123,164)(124,153)(125,172)(126,161)(127,180)(128,169)(129,158)(130,177)(131,166)(132,155)(133,174)(134,163)(135,152)(136,171)(137,160)(138,179)(139,168)(140,157)(141,176)(142,165)(143,154)(144,173)(145,162)(146,151)(147,170)(148,159)(149,178)(150,167)(181,205)(182,194)(184,202)(185,191)(186,210)(187,199)(189,207)(190,196)(192,204)(195,201)(197,209)(200,206)(211,223)(213,231)(214,220)(215,239)(216,228)(218,236)(219,225)(221,233)(224,230)(226,238)(229,235)(234,240) );
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)], [(1,180,16,165),(2,179,17,164),(3,178,18,163),(4,177,19,162),(5,176,20,161),(6,175,21,160),(7,174,22,159),(8,173,23,158),(9,172,24,157),(10,171,25,156),(11,170,26,155),(12,169,27,154),(13,168,28,153),(14,167,29,152),(15,166,30,151),(31,141,46,126),(32,140,47,125),(33,139,48,124),(34,138,49,123),(35,137,50,122),(36,136,51,121),(37,135,52,150),(38,134,53,149),(39,133,54,148),(40,132,55,147),(41,131,56,146),(42,130,57,145),(43,129,58,144),(44,128,59,143),(45,127,60,142),(61,194,76,209),(62,193,77,208),(63,192,78,207),(64,191,79,206),(65,190,80,205),(66,189,81,204),(67,188,82,203),(68,187,83,202),(69,186,84,201),(70,185,85,200),(71,184,86,199),(72,183,87,198),(73,182,88,197),(74,181,89,196),(75,210,90,195),(91,215,106,230),(92,214,107,229),(93,213,108,228),(94,212,109,227),(95,211,110,226),(96,240,111,225),(97,239,112,224),(98,238,113,223),(99,237,114,222),(100,236,115,221),(101,235,116,220),(102,234,117,219),(103,233,118,218),(104,232,119,217),(105,231,120,216)], [(1,232,45,188),(2,213,46,199),(3,224,47,210),(4,235,48,191),(5,216,49,202),(6,227,50,183),(7,238,51,194),(8,219,52,205),(9,230,53,186),(10,211,54,197),(11,222,55,208),(12,233,56,189),(13,214,57,200),(14,225,58,181),(15,236,59,192),(16,217,60,203),(17,228,31,184),(18,239,32,195),(19,220,33,206),(20,231,34,187),(21,212,35,198),(22,223,36,209),(23,234,37,190),(24,215,38,201),(25,226,39,182),(26,237,40,193),(27,218,41,204),(28,229,42,185),(29,240,43,196),(30,221,44,207),(61,159,98,136),(62,170,99,147),(63,151,100,128),(64,162,101,139),(65,173,102,150),(66,154,103,131),(67,165,104,142),(68,176,105,123),(69,157,106,134),(70,168,107,145),(71,179,108,126),(72,160,109,137),(73,171,110,148),(74,152,111,129),(75,163,112,140),(76,174,113,121),(77,155,114,132),(78,166,115,143),(79,177,116,124),(80,158,117,135),(81,169,118,146),(82,180,119,127),(83,161,120,138),(84,172,91,149),(85,153,92,130),(86,164,93,141),(87,175,94,122),(88,156,95,133),(89,167,96,144),(90,178,97,125)], [(1,45),(2,34),(3,53),(4,42),(5,31),(6,50),(7,39),(8,58),(9,47),(10,36),(11,55),(12,44),(13,33),(14,52),(15,41),(16,60),(17,49),(18,38),(19,57),(20,46),(21,35),(22,54),(23,43),(24,32),(25,51),(26,40),(27,59),(28,48),(29,37),(30,56),(61,73),(63,81),(64,70),(65,89),(66,78),(68,86),(69,75),(71,83),(74,80),(76,88),(79,85),(84,90),(91,97),(92,116),(93,105),(95,113),(96,102),(98,110),(100,118),(101,107),(103,115),(106,112),(108,120),(111,117),(121,156),(122,175),(123,164),(124,153),(125,172),(126,161),(127,180),(128,169),(129,158),(130,177),(131,166),(132,155),(133,174),(134,163),(135,152),(136,171),(137,160),(138,179),(139,168),(140,157),(141,176),(142,165),(143,154),(144,173),(145,162),(146,151),(147,170),(148,159),(149,178),(150,167),(181,205),(182,194),(184,202),(185,191),(186,210),(187,199),(189,207),(190,196),(192,204),(195,201),(197,209),(200,206),(211,223),(213,231),(214,220),(215,239),(216,228),(218,236),(219,225),(221,233),(224,230),(226,238),(229,235),(234,240)]])
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 | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 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 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | C4×D5 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | Q8⋊2D5 | C2×S3×D5 | D20⋊S3 | C4×S3×D5 | D10⋊D6 |
| kernel | Dic15⋊13D4 | Dic3×Dic5 | D30⋊4C4 | C3×D10⋊C4 | C5×Dic3⋊C4 | C2×D5×Dic3 | C2×C3⋊D20 | C2×C4×D15 | C3⋊D20 | D10⋊C4 | Dic15 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | D10 | Dic3 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic15⋊13D4 ►in GL6(𝔽61)
| 2 | 15 | 0 | 0 | 0 | 0 |
| 12 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 21 | 0 | 0 | 0 | 0 |
| 37 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 43 | 0 | 0 |
| 0 | 0 | 1 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 31 | 13 | 0 | 0 | 0 | 0 |
| 20 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 6 |
| 0 | 0 | 0 | 0 | 20 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 60 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 20 | 1 |
G:=sub<GL(6,GF(61))| [2,12,0,0,0,0,15,60,0,0,0,0,0,0,43,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,37,0,0,0,0,21,25,0,0,0,0,0,0,18,1,0,0,0,0,43,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[31,20,0,0,0,0,13,30,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,20,0,0,0,0,6,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,60,0,0,0,0,18,18,0,0,0,0,0,0,60,20,0,0,0,0,0,1] >;
Dic15⋊13D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_{13}D_4 % in TeX
G:=Group("Dic15:13D4"); // GroupNames label
G:=SmallGroup(480,472);
// by ID
G=gap.SmallGroup(480,472);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,100,1356,18822]);
// 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=a^11,d*a*d=a^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations