metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊5D4, C3⋊3(C20⋊D4), C15⋊4(C4⋊1D4), C5⋊3(C12⋊3D4), (C5×Dic3)⋊5D4, (C3×Dic5)⋊5D4, C6.165(D4×D5), C10.92(S3×D4), C30.247(C2×D4), C23.25(S3×D5), Dic3⋊1(C5⋊D4), Dic5⋊1(C3⋊D4), (C22×D5).31D6, (C22×C10).61D6, (C22×C6).43D10, (Dic3×Dic5)⋊39C2, (C2×C30).209C23, (C2×Dic5).134D6, (C22×S3).30D10, C2.42(D10⋊D6), (C2×Dic3).127D10, (C22×C30).71C22, (C6×Dic5).121C22, (C22×D15).69C22, (C2×Dic15).143C22, (C10×Dic3).122C22, (C2×C5⋊D4)⋊5S3, (C6×C5⋊D4)⋊5C2, (C2×C3⋊D4)⋊7D5, (C10×C3⋊D4)⋊7C2, C2.42(S3×C5⋊D4), C6.67(C2×C5⋊D4), C2.45(D5×C3⋊D4), (C2×C5⋊D12)⋊14C2, (C2×C15⋊D4)⋊14C2, (C2×C15⋊7D4)⋊18C2, (C2×C3⋊D20)⋊14C2, C10.69(C2×C3⋊D4), (D5×C2×C6).54C22, C22.238(C2×S3×D5), (S3×C2×C10).55C22, (C2×C6).221(C22×D5), (C2×C10).221(C22×S3), SmallGroup(480,643)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊5D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a11, bc=cb, dbd=a15b, dcd=c-1 >
Subgroups: 1308 in 216 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, D15, C30, C30, C4⋊1D4, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C4×Dic3, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, S3×C10, D30, C2×C30, C2×C30, C4×Dic5, C2×D20, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C12⋊3D4, C15⋊D4, C3⋊D20, C5⋊D12, C6×Dic5, C3×C5⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, C15⋊7D4, D5×C2×C6, S3×C2×C10, C22×D15, C22×C30, C20⋊D4, Dic3×Dic5, C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, C6×C5⋊D4, C10×C3⋊D4, C2×C15⋊7D4, Dic15⋊5D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4⋊1D4, C5⋊D4, C22×D5, S3×D4, C2×C3⋊D4, S3×D5, D4×D5, C2×C5⋊D4, C12⋊3D4, C2×S3×D5, C20⋊D4, D5×C3⋊D4, S3×C5⋊D4, D10⋊D6, Dic15⋊5D4
►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 91 16 106)(2 120 17 105)(3 119 18 104)(4 118 19 103)(5 117 20 102)(6 116 21 101)(7 115 22 100)(8 114 23 99)(9 113 24 98)(10 112 25 97)(11 111 26 96)(12 110 27 95)(13 109 28 94)(14 108 29 93)(15 107 30 92)(31 82 46 67)(32 81 47 66)(33 80 48 65)(34 79 49 64)(35 78 50 63)(36 77 51 62)(37 76 52 61)(38 75 53 90)(39 74 54 89)(40 73 55 88)(41 72 56 87)(42 71 57 86)(43 70 58 85)(44 69 59 84)(45 68 60 83)(121 223 136 238)(122 222 137 237)(123 221 138 236)(124 220 139 235)(125 219 140 234)(126 218 141 233)(127 217 142 232)(128 216 143 231)(129 215 144 230)(130 214 145 229)(131 213 146 228)(132 212 147 227)(133 211 148 226)(134 240 149 225)(135 239 150 224)(151 200 166 185)(152 199 167 184)(153 198 168 183)(154 197 169 182)(155 196 170 181)(156 195 171 210)(157 194 172 209)(158 193 173 208)(159 192 174 207)(160 191 175 206)(161 190 176 205)(162 189 177 204)(163 188 178 203)(164 187 179 202)(165 186 180 201)
(1 226 37 195)(2 237 38 206)(3 218 39 187)(4 229 40 198)(5 240 41 209)(6 221 42 190)(7 232 43 201)(8 213 44 182)(9 224 45 193)(10 235 46 204)(11 216 47 185)(12 227 48 196)(13 238 49 207)(14 219 50 188)(15 230 51 199)(16 211 52 210)(17 222 53 191)(18 233 54 202)(19 214 55 183)(20 225 56 194)(21 236 57 205)(22 217 58 186)(23 228 59 197)(24 239 60 208)(25 220 31 189)(26 231 32 200)(27 212 33 181)(28 223 34 192)(29 234 35 203)(30 215 36 184)(61 156 106 148)(62 167 107 129)(63 178 108 140)(64 159 109 121)(65 170 110 132)(66 151 111 143)(67 162 112 124)(68 173 113 135)(69 154 114 146)(70 165 115 127)(71 176 116 138)(72 157 117 149)(73 168 118 130)(74 179 119 141)(75 160 120 122)(76 171 91 133)(77 152 92 144)(78 163 93 125)(79 174 94 136)(80 155 95 147)(81 166 96 128)(82 177 97 139)(83 158 98 150)(84 169 99 131)(85 180 100 142)(86 161 101 123)(87 172 102 134)(88 153 103 145)(89 164 104 126)(90 175 105 137)
(1 148)(2 129)(3 140)(4 121)(5 132)(6 143)(7 124)(8 135)(9 146)(10 127)(11 138)(12 149)(13 130)(14 141)(15 122)(16 133)(17 144)(18 125)(19 136)(20 147)(21 128)(22 139)(23 150)(24 131)(25 142)(26 123)(27 134)(28 145)(29 126)(30 137)(31 180)(32 161)(33 172)(34 153)(35 164)(36 175)(37 156)(38 167)(39 178)(40 159)(41 170)(42 151)(43 162)(44 173)(45 154)(46 165)(47 176)(48 157)(49 168)(50 179)(51 160)(52 171)(53 152)(54 163)(55 174)(56 155)(57 166)(58 177)(59 158)(60 169)(61 195)(62 206)(63 187)(64 198)(65 209)(66 190)(67 201)(68 182)(69 193)(70 204)(71 185)(72 196)(73 207)(74 188)(75 199)(76 210)(77 191)(78 202)(79 183)(80 194)(81 205)(82 186)(83 197)(84 208)(85 189)(86 200)(87 181)(88 192)(89 203)(90 184)(91 211)(92 222)(93 233)(94 214)(95 225)(96 236)(97 217)(98 228)(99 239)(100 220)(101 231)(102 212)(103 223)(104 234)(105 215)(106 226)(107 237)(108 218)(109 229)(110 240)(111 221)(112 232)(113 213)(114 224)(115 235)(116 216)(117 227)(118 238)(119 219)(120 230)
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), (1,91,16,106)(2,120,17,105)(3,119,18,104)(4,118,19,103)(5,117,20,102)(6,116,21,101)(7,115,22,100)(8,114,23,99)(9,113,24,98)(10,112,25,97)(11,111,26,96)(12,110,27,95)(13,109,28,94)(14,108,29,93)(15,107,30,92)(31,82,46,67)(32,81,47,66)(33,80,48,65)(34,79,49,64)(35,78,50,63)(36,77,51,62)(37,76,52,61)(38,75,53,90)(39,74,54,89)(40,73,55,88)(41,72,56,87)(42,71,57,86)(43,70,58,85)(44,69,59,84)(45,68,60,83)(121,223,136,238)(122,222,137,237)(123,221,138,236)(124,220,139,235)(125,219,140,234)(126,218,141,233)(127,217,142,232)(128,216,143,231)(129,215,144,230)(130,214,145,229)(131,213,146,228)(132,212,147,227)(133,211,148,226)(134,240,149,225)(135,239,150,224)(151,200,166,185)(152,199,167,184)(153,198,168,183)(154,197,169,182)(155,196,170,181)(156,195,171,210)(157,194,172,209)(158,193,173,208)(159,192,174,207)(160,191,175,206)(161,190,176,205)(162,189,177,204)(163,188,178,203)(164,187,179,202)(165,186,180,201), (1,226,37,195)(2,237,38,206)(3,218,39,187)(4,229,40,198)(5,240,41,209)(6,221,42,190)(7,232,43,201)(8,213,44,182)(9,224,45,193)(10,235,46,204)(11,216,47,185)(12,227,48,196)(13,238,49,207)(14,219,50,188)(15,230,51,199)(16,211,52,210)(17,222,53,191)(18,233,54,202)(19,214,55,183)(20,225,56,194)(21,236,57,205)(22,217,58,186)(23,228,59,197)(24,239,60,208)(25,220,31,189)(26,231,32,200)(27,212,33,181)(28,223,34,192)(29,234,35,203)(30,215,36,184)(61,156,106,148)(62,167,107,129)(63,178,108,140)(64,159,109,121)(65,170,110,132)(66,151,111,143)(67,162,112,124)(68,173,113,135)(69,154,114,146)(70,165,115,127)(71,176,116,138)(72,157,117,149)(73,168,118,130)(74,179,119,141)(75,160,120,122)(76,171,91,133)(77,152,92,144)(78,163,93,125)(79,174,94,136)(80,155,95,147)(81,166,96,128)(82,177,97,139)(83,158,98,150)(84,169,99,131)(85,180,100,142)(86,161,101,123)(87,172,102,134)(88,153,103,145)(89,164,104,126)(90,175,105,137), (1,148)(2,129)(3,140)(4,121)(5,132)(6,143)(7,124)(8,135)(9,146)(10,127)(11,138)(12,149)(13,130)(14,141)(15,122)(16,133)(17,144)(18,125)(19,136)(20,147)(21,128)(22,139)(23,150)(24,131)(25,142)(26,123)(27,134)(28,145)(29,126)(30,137)(31,180)(32,161)(33,172)(34,153)(35,164)(36,175)(37,156)(38,167)(39,178)(40,159)(41,170)(42,151)(43,162)(44,173)(45,154)(46,165)(47,176)(48,157)(49,168)(50,179)(51,160)(52,171)(53,152)(54,163)(55,174)(56,155)(57,166)(58,177)(59,158)(60,169)(61,195)(62,206)(63,187)(64,198)(65,209)(66,190)(67,201)(68,182)(69,193)(70,204)(71,185)(72,196)(73,207)(74,188)(75,199)(76,210)(77,191)(78,202)(79,183)(80,194)(81,205)(82,186)(83,197)(84,208)(85,189)(86,200)(87,181)(88,192)(89,203)(90,184)(91,211)(92,222)(93,233)(94,214)(95,225)(96,236)(97,217)(98,228)(99,239)(100,220)(101,231)(102,212)(103,223)(104,234)(105,215)(106,226)(107,237)(108,218)(109,229)(110,240)(111,221)(112,232)(113,213)(114,224)(115,235)(116,216)(117,227)(118,238)(119,219)(120,230)>;
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), (1,91,16,106)(2,120,17,105)(3,119,18,104)(4,118,19,103)(5,117,20,102)(6,116,21,101)(7,115,22,100)(8,114,23,99)(9,113,24,98)(10,112,25,97)(11,111,26,96)(12,110,27,95)(13,109,28,94)(14,108,29,93)(15,107,30,92)(31,82,46,67)(32,81,47,66)(33,80,48,65)(34,79,49,64)(35,78,50,63)(36,77,51,62)(37,76,52,61)(38,75,53,90)(39,74,54,89)(40,73,55,88)(41,72,56,87)(42,71,57,86)(43,70,58,85)(44,69,59,84)(45,68,60,83)(121,223,136,238)(122,222,137,237)(123,221,138,236)(124,220,139,235)(125,219,140,234)(126,218,141,233)(127,217,142,232)(128,216,143,231)(129,215,144,230)(130,214,145,229)(131,213,146,228)(132,212,147,227)(133,211,148,226)(134,240,149,225)(135,239,150,224)(151,200,166,185)(152,199,167,184)(153,198,168,183)(154,197,169,182)(155,196,170,181)(156,195,171,210)(157,194,172,209)(158,193,173,208)(159,192,174,207)(160,191,175,206)(161,190,176,205)(162,189,177,204)(163,188,178,203)(164,187,179,202)(165,186,180,201), (1,226,37,195)(2,237,38,206)(3,218,39,187)(4,229,40,198)(5,240,41,209)(6,221,42,190)(7,232,43,201)(8,213,44,182)(9,224,45,193)(10,235,46,204)(11,216,47,185)(12,227,48,196)(13,238,49,207)(14,219,50,188)(15,230,51,199)(16,211,52,210)(17,222,53,191)(18,233,54,202)(19,214,55,183)(20,225,56,194)(21,236,57,205)(22,217,58,186)(23,228,59,197)(24,239,60,208)(25,220,31,189)(26,231,32,200)(27,212,33,181)(28,223,34,192)(29,234,35,203)(30,215,36,184)(61,156,106,148)(62,167,107,129)(63,178,108,140)(64,159,109,121)(65,170,110,132)(66,151,111,143)(67,162,112,124)(68,173,113,135)(69,154,114,146)(70,165,115,127)(71,176,116,138)(72,157,117,149)(73,168,118,130)(74,179,119,141)(75,160,120,122)(76,171,91,133)(77,152,92,144)(78,163,93,125)(79,174,94,136)(80,155,95,147)(81,166,96,128)(82,177,97,139)(83,158,98,150)(84,169,99,131)(85,180,100,142)(86,161,101,123)(87,172,102,134)(88,153,103,145)(89,164,104,126)(90,175,105,137), (1,148)(2,129)(3,140)(4,121)(5,132)(6,143)(7,124)(8,135)(9,146)(10,127)(11,138)(12,149)(13,130)(14,141)(15,122)(16,133)(17,144)(18,125)(19,136)(20,147)(21,128)(22,139)(23,150)(24,131)(25,142)(26,123)(27,134)(28,145)(29,126)(30,137)(31,180)(32,161)(33,172)(34,153)(35,164)(36,175)(37,156)(38,167)(39,178)(40,159)(41,170)(42,151)(43,162)(44,173)(45,154)(46,165)(47,176)(48,157)(49,168)(50,179)(51,160)(52,171)(53,152)(54,163)(55,174)(56,155)(57,166)(58,177)(59,158)(60,169)(61,195)(62,206)(63,187)(64,198)(65,209)(66,190)(67,201)(68,182)(69,193)(70,204)(71,185)(72,196)(73,207)(74,188)(75,199)(76,210)(77,191)(78,202)(79,183)(80,194)(81,205)(82,186)(83,197)(84,208)(85,189)(86,200)(87,181)(88,192)(89,203)(90,184)(91,211)(92,222)(93,233)(94,214)(95,225)(96,236)(97,217)(98,228)(99,239)(100,220)(101,231)(102,212)(103,223)(104,234)(105,215)(106,226)(107,237)(108,218)(109,229)(110,240)(111,221)(112,232)(113,213)(114,224)(115,235)(116,216)(117,227)(118,238)(119,219)(120,230) );
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)], [(1,91,16,106),(2,120,17,105),(3,119,18,104),(4,118,19,103),(5,117,20,102),(6,116,21,101),(7,115,22,100),(8,114,23,99),(9,113,24,98),(10,112,25,97),(11,111,26,96),(12,110,27,95),(13,109,28,94),(14,108,29,93),(15,107,30,92),(31,82,46,67),(32,81,47,66),(33,80,48,65),(34,79,49,64),(35,78,50,63),(36,77,51,62),(37,76,52,61),(38,75,53,90),(39,74,54,89),(40,73,55,88),(41,72,56,87),(42,71,57,86),(43,70,58,85),(44,69,59,84),(45,68,60,83),(121,223,136,238),(122,222,137,237),(123,221,138,236),(124,220,139,235),(125,219,140,234),(126,218,141,233),(127,217,142,232),(128,216,143,231),(129,215,144,230),(130,214,145,229),(131,213,146,228),(132,212,147,227),(133,211,148,226),(134,240,149,225),(135,239,150,224),(151,200,166,185),(152,199,167,184),(153,198,168,183),(154,197,169,182),(155,196,170,181),(156,195,171,210),(157,194,172,209),(158,193,173,208),(159,192,174,207),(160,191,175,206),(161,190,176,205),(162,189,177,204),(163,188,178,203),(164,187,179,202),(165,186,180,201)], [(1,226,37,195),(2,237,38,206),(3,218,39,187),(4,229,40,198),(5,240,41,209),(6,221,42,190),(7,232,43,201),(8,213,44,182),(9,224,45,193),(10,235,46,204),(11,216,47,185),(12,227,48,196),(13,238,49,207),(14,219,50,188),(15,230,51,199),(16,211,52,210),(17,222,53,191),(18,233,54,202),(19,214,55,183),(20,225,56,194),(21,236,57,205),(22,217,58,186),(23,228,59,197),(24,239,60,208),(25,220,31,189),(26,231,32,200),(27,212,33,181),(28,223,34,192),(29,234,35,203),(30,215,36,184),(61,156,106,148),(62,167,107,129),(63,178,108,140),(64,159,109,121),(65,170,110,132),(66,151,111,143),(67,162,112,124),(68,173,113,135),(69,154,114,146),(70,165,115,127),(71,176,116,138),(72,157,117,149),(73,168,118,130),(74,179,119,141),(75,160,120,122),(76,171,91,133),(77,152,92,144),(78,163,93,125),(79,174,94,136),(80,155,95,147),(81,166,96,128),(82,177,97,139),(83,158,98,150),(84,169,99,131),(85,180,100,142),(86,161,101,123),(87,172,102,134),(88,153,103,145),(89,164,104,126),(90,175,105,137)], [(1,148),(2,129),(3,140),(4,121),(5,132),(6,143),(7,124),(8,135),(9,146),(10,127),(11,138),(12,149),(13,130),(14,141),(15,122),(16,133),(17,144),(18,125),(19,136),(20,147),(21,128),(22,139),(23,150),(24,131),(25,142),(26,123),(27,134),(28,145),(29,126),(30,137),(31,180),(32,161),(33,172),(34,153),(35,164),(36,175),(37,156),(38,167),(39,178),(40,159),(41,170),(42,151),(43,162),(44,173),(45,154),(46,165),(47,176),(48,157),(49,168),(50,179),(51,160),(52,171),(53,152),(54,163),(55,174),(56,155),(57,166),(58,177),(59,158),(60,169),(61,195),(62,206),(63,187),(64,198),(65,209),(66,190),(67,201),(68,182),(69,193),(70,204),(71,185),(72,196),(73,207),(74,188),(75,199),(76,210),(77,191),(78,202),(79,183),(80,194),(81,205),(82,186),(83,197),(84,208),(85,189),(86,200),(87,181),(88,192),(89,203),(90,184),(91,211),(92,222),(93,233),(94,214),(95,225),(96,236),(97,217),(98,228),(99,239),(100,220),(101,231),(102,212),(103,223),(104,234),(105,215),(106,226),(107,237),(108,218),(109,229),(110,240),(111,221),(112,232),(113,213),(114,224),(115,235),(116,216),(117,227),(118,238),(119,219),(120,230)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 20 | 60 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | D5×C3⋊D4 | S3×C5⋊D4 | D10⋊D6 |
| kernel | Dic15⋊5D4 | Dic3×Dic5 | C2×C15⋊D4 | C2×C3⋊D20 | C2×C5⋊D12 | C6×C5⋊D4 | C10×C3⋊D4 | C2×C15⋊7D4 | C2×C5⋊D4 | C5×Dic3 | C3×Dic5 | Dic15 | C2×C3⋊D4 | C2×Dic5 | C22×D5 | C22×C10 | C2×Dic3 | C22×S3 | C22×C6 | Dic5 | Dic3 | C10 | C23 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of Dic15⋊5D4 ►in GL4(𝔽61) generated by
| 60 | 9 | 0 | 0 |
| 20 | 2 | 0 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 0 | 19 | 43 |
| 27 | 21 | 0 | 0 |
| 3 | 34 | 0 | 0 |
| 0 | 0 | 8 | 14 |
| 0 | 0 | 52 | 53 |
| 34 | 40 | 0 | 0 |
| 58 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 41 | 60 | 0 | 0 |
| 0 | 0 | 30 | 45 |
| 0 | 0 | 60 | 31 |
G:=sub<GL(4,GF(61))| [60,20,0,0,9,2,0,0,0,0,1,19,0,0,60,43],[27,3,0,0,21,34,0,0,0,0,8,52,0,0,14,53],[34,58,0,0,40,27,0,0,0,0,1,0,0,0,0,1],[1,41,0,0,0,60,0,0,0,0,30,60,0,0,45,31] >;
Dic15⋊5D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_5D_4 % in TeX
G:=Group("Dic15:5D4"); // GroupNames label
G:=SmallGroup(480,643);
// by ID
G=gap.SmallGroup(480,643);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,303,1356,18822]);
// 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^11,b*c=c*b,d*b*d=a^15*b,d*c*d=c^-1>;
// generators/relations