metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊7D4, D20⋊8Dic3, C12⋊3(C4×D5), C15⋊13(C4×D4), C5⋊3(D4×Dic3), C60⋊20(C2×C4), C4⋊2(D5×Dic3), C6.40(D4×D5), (C3×D20)⋊12C4, C4⋊Dic3⋊13D5, (C6×D20).9C2, C20⋊5(C2×Dic3), C30.52(C2×D4), C10.41(S3×D4), C3⋊5(D20⋊8C4), D10⋊3(C2×Dic3), (C2×D20).10S3, (C2×C20).126D6, (C4×Dic15)⋊25C2, C30.73(C4○D4), (C2×C12).127D10, C2.1(C20⋊D6), (C22×D5).54D6, C2.5(D20⋊S3), D10⋊Dic3⋊11C2, (C2×C30).124C23, (C2×C60).200C22, C30.130(C22×C4), C6.36(Q8⋊2D5), C10.14(D4⋊2S3), (C2×Dic3).107D10, C10.26(C22×Dic3), (C10×Dic3).78C22, (C2×Dic15).206C22, C6.89(C2×C4×D5), (C6×D5)⋊4(C2×C4), (C2×D5×Dic3)⋊9C2, C2.14(C2×D5×Dic3), C22.58(C2×S3×D5), (C5×C4⋊Dic3)⋊10C2, (C2×C4).210(S3×D5), (D5×C2×C6).24C22, (C2×C6).136(C22×D5), (C2×C10).136(C22×S3), SmallGroup(480,510)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊7D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, ac=ca, dad=a19, bc=cb, bd=db, dcd=c-1 >
Subgroups: 892 in 188 conjugacy classes, 70 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×4], C10 [×3], Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×8], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×Dic3 [×2], C2×Dic3 [×6], C2×C12, C3×D4 [×4], C22×C6 [×2], C3×D5 [×4], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3 [×2], C6×D4, C5×Dic3 [×2], Dic15 [×2], Dic15, C60 [×2], C6×D5 [×4], C6×D5 [×4], C2×C30, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, D4×Dic3, D5×Dic3 [×4], C3×D20 [×4], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], D20⋊8C4, D10⋊Dic3 [×2], C5×C4⋊Dic3, C4×Dic15, C2×D5×Dic3 [×2], C6×D20, Dic15⋊7D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C2×Dic3 [×6], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×D4, D4⋊2S3, C22×Dic3, S3×D5, C2×C4×D5, D4×D5, Q8⋊2D5, D4×Dic3, D5×Dic3 [×2], C2×S3×D5, D20⋊8C4, D20⋊S3, C20⋊D6, C2×D5×Dic3, Dic15⋊7D4
►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 96 46 111)(32 95 47 110)(33 94 48 109)(34 93 49 108)(35 92 50 107)(36 91 51 106)(37 120 52 105)(38 119 53 104)(39 118 54 103)(40 117 55 102)(41 116 56 101)(42 115 57 100)(43 114 58 99)(44 113 59 98)(45 112 60 97)(61 196 76 181)(62 195 77 210)(63 194 78 209)(64 193 79 208)(65 192 80 207)(66 191 81 206)(67 190 82 205)(68 189 83 204)(69 188 84 203)(70 187 85 202)(71 186 86 201)(72 185 87 200)(73 184 88 199)(74 183 89 198)(75 182 90 197)(121 220 136 235)(122 219 137 234)(123 218 138 233)(124 217 139 232)(125 216 140 231)(126 215 141 230)(127 214 142 229)(128 213 143 228)(129 212 144 227)(130 211 145 226)(131 240 146 225)(132 239 147 224)(133 238 148 223)(134 237 149 222)(135 236 150 221)
(1 210 50 231)(2 181 51 232)(3 182 52 233)(4 183 53 234)(5 184 54 235)(6 185 55 236)(7 186 56 237)(8 187 57 238)(9 188 58 239)(10 189 59 240)(11 190 60 211)(12 191 31 212)(13 192 32 213)(14 193 33 214)(15 194 34 215)(16 195 35 216)(17 196 36 217)(18 197 37 218)(19 198 38 219)(20 199 39 220)(21 200 40 221)(22 201 41 222)(23 202 42 223)(24 203 43 224)(25 204 44 225)(26 205 45 226)(27 206 46 227)(28 207 47 228)(29 208 48 229)(30 209 49 230)(61 106 124 179)(62 107 125 180)(63 108 126 151)(64 109 127 152)(65 110 128 153)(66 111 129 154)(67 112 130 155)(68 113 131 156)(69 114 132 157)(70 115 133 158)(71 116 134 159)(72 117 135 160)(73 118 136 161)(74 119 137 162)(75 120 138 163)(76 91 139 164)(77 92 140 165)(78 93 141 166)(79 94 142 167)(80 95 143 168)(81 96 144 169)(82 97 145 170)(83 98 146 171)(84 99 147 172)(85 100 148 173)(86 101 149 174)(87 102 150 175)(88 103 121 176)(89 104 122 177)(90 105 123 178)
(1 210)(2 199)(3 188)(4 207)(5 196)(6 185)(7 204)(8 193)(9 182)(10 201)(11 190)(12 209)(13 198)(14 187)(15 206)(16 195)(17 184)(18 203)(19 192)(20 181)(21 200)(22 189)(23 208)(24 197)(25 186)(26 205)(27 194)(28 183)(29 202)(30 191)(31 230)(32 219)(33 238)(34 227)(35 216)(36 235)(37 224)(38 213)(39 232)(40 221)(41 240)(42 229)(43 218)(44 237)(45 226)(46 215)(47 234)(48 223)(49 212)(50 231)(51 220)(52 239)(53 228)(54 217)(55 236)(56 225)(57 214)(58 233)(59 222)(60 211)(61 161)(62 180)(63 169)(64 158)(65 177)(66 166)(67 155)(68 174)(69 163)(70 152)(71 171)(72 160)(73 179)(74 168)(75 157)(76 176)(77 165)(78 154)(79 173)(80 162)(81 151)(82 170)(83 159)(84 178)(85 167)(86 156)(87 175)(88 164)(89 153)(90 172)(91 121)(92 140)(93 129)(94 148)(95 137)(96 126)(97 145)(98 134)(99 123)(100 142)(101 131)(102 150)(103 139)(104 128)(105 147)(106 136)(107 125)(108 144)(109 133)(110 122)(111 141)(112 130)(113 149)(114 138)(115 127)(116 146)(117 135)(118 124)(119 143)(120 132)
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)(31,96,46,111)(32,95,47,110)(33,94,48,109)(34,93,49,108)(35,92,50,107)(36,91,51,106)(37,120,52,105)(38,119,53,104)(39,118,54,103)(40,117,55,102)(41,116,56,101)(42,115,57,100)(43,114,58,99)(44,113,59,98)(45,112,60,97)(61,196,76,181)(62,195,77,210)(63,194,78,209)(64,193,79,208)(65,192,80,207)(66,191,81,206)(67,190,82,205)(68,189,83,204)(69,188,84,203)(70,187,85,202)(71,186,86,201)(72,185,87,200)(73,184,88,199)(74,183,89,198)(75,182,90,197)(121,220,136,235)(122,219,137,234)(123,218,138,233)(124,217,139,232)(125,216,140,231)(126,215,141,230)(127,214,142,229)(128,213,143,228)(129,212,144,227)(130,211,145,226)(131,240,146,225)(132,239,147,224)(133,238,148,223)(134,237,149,222)(135,236,150,221), (1,210,50,231)(2,181,51,232)(3,182,52,233)(4,183,53,234)(5,184,54,235)(6,185,55,236)(7,186,56,237)(8,187,57,238)(9,188,58,239)(10,189,59,240)(11,190,60,211)(12,191,31,212)(13,192,32,213)(14,193,33,214)(15,194,34,215)(16,195,35,216)(17,196,36,217)(18,197,37,218)(19,198,38,219)(20,199,39,220)(21,200,40,221)(22,201,41,222)(23,202,42,223)(24,203,43,224)(25,204,44,225)(26,205,45,226)(27,206,46,227)(28,207,47,228)(29,208,48,229)(30,209,49,230)(61,106,124,179)(62,107,125,180)(63,108,126,151)(64,109,127,152)(65,110,128,153)(66,111,129,154)(67,112,130,155)(68,113,131,156)(69,114,132,157)(70,115,133,158)(71,116,134,159)(72,117,135,160)(73,118,136,161)(74,119,137,162)(75,120,138,163)(76,91,139,164)(77,92,140,165)(78,93,141,166)(79,94,142,167)(80,95,143,168)(81,96,144,169)(82,97,145,170)(83,98,146,171)(84,99,147,172)(85,100,148,173)(86,101,149,174)(87,102,150,175)(88,103,121,176)(89,104,122,177)(90,105,123,178), (1,210)(2,199)(3,188)(4,207)(5,196)(6,185)(7,204)(8,193)(9,182)(10,201)(11,190)(12,209)(13,198)(14,187)(15,206)(16,195)(17,184)(18,203)(19,192)(20,181)(21,200)(22,189)(23,208)(24,197)(25,186)(26,205)(27,194)(28,183)(29,202)(30,191)(31,230)(32,219)(33,238)(34,227)(35,216)(36,235)(37,224)(38,213)(39,232)(40,221)(41,240)(42,229)(43,218)(44,237)(45,226)(46,215)(47,234)(48,223)(49,212)(50,231)(51,220)(52,239)(53,228)(54,217)(55,236)(56,225)(57,214)(58,233)(59,222)(60,211)(61,161)(62,180)(63,169)(64,158)(65,177)(66,166)(67,155)(68,174)(69,163)(70,152)(71,171)(72,160)(73,179)(74,168)(75,157)(76,176)(77,165)(78,154)(79,173)(80,162)(81,151)(82,170)(83,159)(84,178)(85,167)(86,156)(87,175)(88,164)(89,153)(90,172)(91,121)(92,140)(93,129)(94,148)(95,137)(96,126)(97,145)(98,134)(99,123)(100,142)(101,131)(102,150)(103,139)(104,128)(105,147)(106,136)(107,125)(108,144)(109,133)(110,122)(111,141)(112,130)(113,149)(114,138)(115,127)(116,146)(117,135)(118,124)(119,143)(120,132)>;
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)(31,96,46,111)(32,95,47,110)(33,94,48,109)(34,93,49,108)(35,92,50,107)(36,91,51,106)(37,120,52,105)(38,119,53,104)(39,118,54,103)(40,117,55,102)(41,116,56,101)(42,115,57,100)(43,114,58,99)(44,113,59,98)(45,112,60,97)(61,196,76,181)(62,195,77,210)(63,194,78,209)(64,193,79,208)(65,192,80,207)(66,191,81,206)(67,190,82,205)(68,189,83,204)(69,188,84,203)(70,187,85,202)(71,186,86,201)(72,185,87,200)(73,184,88,199)(74,183,89,198)(75,182,90,197)(121,220,136,235)(122,219,137,234)(123,218,138,233)(124,217,139,232)(125,216,140,231)(126,215,141,230)(127,214,142,229)(128,213,143,228)(129,212,144,227)(130,211,145,226)(131,240,146,225)(132,239,147,224)(133,238,148,223)(134,237,149,222)(135,236,150,221), (1,210,50,231)(2,181,51,232)(3,182,52,233)(4,183,53,234)(5,184,54,235)(6,185,55,236)(7,186,56,237)(8,187,57,238)(9,188,58,239)(10,189,59,240)(11,190,60,211)(12,191,31,212)(13,192,32,213)(14,193,33,214)(15,194,34,215)(16,195,35,216)(17,196,36,217)(18,197,37,218)(19,198,38,219)(20,199,39,220)(21,200,40,221)(22,201,41,222)(23,202,42,223)(24,203,43,224)(25,204,44,225)(26,205,45,226)(27,206,46,227)(28,207,47,228)(29,208,48,229)(30,209,49,230)(61,106,124,179)(62,107,125,180)(63,108,126,151)(64,109,127,152)(65,110,128,153)(66,111,129,154)(67,112,130,155)(68,113,131,156)(69,114,132,157)(70,115,133,158)(71,116,134,159)(72,117,135,160)(73,118,136,161)(74,119,137,162)(75,120,138,163)(76,91,139,164)(77,92,140,165)(78,93,141,166)(79,94,142,167)(80,95,143,168)(81,96,144,169)(82,97,145,170)(83,98,146,171)(84,99,147,172)(85,100,148,173)(86,101,149,174)(87,102,150,175)(88,103,121,176)(89,104,122,177)(90,105,123,178), (1,210)(2,199)(3,188)(4,207)(5,196)(6,185)(7,204)(8,193)(9,182)(10,201)(11,190)(12,209)(13,198)(14,187)(15,206)(16,195)(17,184)(18,203)(19,192)(20,181)(21,200)(22,189)(23,208)(24,197)(25,186)(26,205)(27,194)(28,183)(29,202)(30,191)(31,230)(32,219)(33,238)(34,227)(35,216)(36,235)(37,224)(38,213)(39,232)(40,221)(41,240)(42,229)(43,218)(44,237)(45,226)(46,215)(47,234)(48,223)(49,212)(50,231)(51,220)(52,239)(53,228)(54,217)(55,236)(56,225)(57,214)(58,233)(59,222)(60,211)(61,161)(62,180)(63,169)(64,158)(65,177)(66,166)(67,155)(68,174)(69,163)(70,152)(71,171)(72,160)(73,179)(74,168)(75,157)(76,176)(77,165)(78,154)(79,173)(80,162)(81,151)(82,170)(83,159)(84,178)(85,167)(86,156)(87,175)(88,164)(89,153)(90,172)(91,121)(92,140)(93,129)(94,148)(95,137)(96,126)(97,145)(98,134)(99,123)(100,142)(101,131)(102,150)(103,139)(104,128)(105,147)(106,136)(107,125)(108,144)(109,133)(110,122)(111,141)(112,130)(113,149)(114,138)(115,127)(116,146)(117,135)(118,124)(119,143)(120,132) );
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),(31,96,46,111),(32,95,47,110),(33,94,48,109),(34,93,49,108),(35,92,50,107),(36,91,51,106),(37,120,52,105),(38,119,53,104),(39,118,54,103),(40,117,55,102),(41,116,56,101),(42,115,57,100),(43,114,58,99),(44,113,59,98),(45,112,60,97),(61,196,76,181),(62,195,77,210),(63,194,78,209),(64,193,79,208),(65,192,80,207),(66,191,81,206),(67,190,82,205),(68,189,83,204),(69,188,84,203),(70,187,85,202),(71,186,86,201),(72,185,87,200),(73,184,88,199),(74,183,89,198),(75,182,90,197),(121,220,136,235),(122,219,137,234),(123,218,138,233),(124,217,139,232),(125,216,140,231),(126,215,141,230),(127,214,142,229),(128,213,143,228),(129,212,144,227),(130,211,145,226),(131,240,146,225),(132,239,147,224),(133,238,148,223),(134,237,149,222),(135,236,150,221)], [(1,210,50,231),(2,181,51,232),(3,182,52,233),(4,183,53,234),(5,184,54,235),(6,185,55,236),(7,186,56,237),(8,187,57,238),(9,188,58,239),(10,189,59,240),(11,190,60,211),(12,191,31,212),(13,192,32,213),(14,193,33,214),(15,194,34,215),(16,195,35,216),(17,196,36,217),(18,197,37,218),(19,198,38,219),(20,199,39,220),(21,200,40,221),(22,201,41,222),(23,202,42,223),(24,203,43,224),(25,204,44,225),(26,205,45,226),(27,206,46,227),(28,207,47,228),(29,208,48,229),(30,209,49,230),(61,106,124,179),(62,107,125,180),(63,108,126,151),(64,109,127,152),(65,110,128,153),(66,111,129,154),(67,112,130,155),(68,113,131,156),(69,114,132,157),(70,115,133,158),(71,116,134,159),(72,117,135,160),(73,118,136,161),(74,119,137,162),(75,120,138,163),(76,91,139,164),(77,92,140,165),(78,93,141,166),(79,94,142,167),(80,95,143,168),(81,96,144,169),(82,97,145,170),(83,98,146,171),(84,99,147,172),(85,100,148,173),(86,101,149,174),(87,102,150,175),(88,103,121,176),(89,104,122,177),(90,105,123,178)], [(1,210),(2,199),(3,188),(4,207),(5,196),(6,185),(7,204),(8,193),(9,182),(10,201),(11,190),(12,209),(13,198),(14,187),(15,206),(16,195),(17,184),(18,203),(19,192),(20,181),(21,200),(22,189),(23,208),(24,197),(25,186),(26,205),(27,194),(28,183),(29,202),(30,191),(31,230),(32,219),(33,238),(34,227),(35,216),(36,235),(37,224),(38,213),(39,232),(40,221),(41,240),(42,229),(43,218),(44,237),(45,226),(46,215),(47,234),(48,223),(49,212),(50,231),(51,220),(52,239),(53,228),(54,217),(55,236),(56,225),(57,214),(58,233),(59,222),(60,211),(61,161),(62,180),(63,169),(64,158),(65,177),(66,166),(67,155),(68,174),(69,163),(70,152),(71,171),(72,160),(73,179),(74,168),(75,157),(76,176),(77,165),(78,154),(79,173),(80,162),(81,151),(82,170),(83,159),(84,178),(85,167),(86,156),(87,175),(88,164),(89,153),(90,172),(91,121),(92,140),(93,129),(94,148),(95,137),(96,126),(97,145),(98,134),(99,123),(100,142),(101,131),(102,150),(103,139),(104,128),(105,147),(106,136),(107,125),(108,144),(109,133),(110,122),(111,141),(112,130),(113,149),(114,138),(115,127),(116,146),(117,135),(118,124),(119,143),(120,132)])
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 | 12A | 12B | 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 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4 | S3 | D4 | D5 | Dic3 | D6 | D6 | C4○D4 | D10 | D10 | C4×D5 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | Q8⋊2D5 | D5×Dic3 | C2×S3×D5 | D20⋊S3 | C20⋊D6 |
| kernel | Dic15⋊7D4 | D10⋊Dic3 | C5×C4⋊Dic3 | C4×Dic15 | C2×D5×Dic3 | C6×D20 | C3×D20 | C2×D20 | Dic15 | C4⋊Dic3 | D20 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C12 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic15⋊7D4 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 4 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 37 |
| 0 | 0 | 0 | 0 | 10 | 27 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 46 | 0 | 0 |
| 0 | 0 | 53 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 44 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[11,4,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,34,10,0,0,0,0,37,27],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,53,0,0,0,0,46,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,15,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
Dic15⋊7D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_7D_4 % in TeX
G:=Group("Dic15:7D4"); // GroupNames label
G:=SmallGroup(480,510);
// by ID
G=gap.SmallGroup(480,510);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations