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G = Dic1514D4order 480 = 25·3·5

4th semidirect product of Dic15 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1514D4, C158(C4×D4), D62(C4×D5), D6⋊C410D5, C5⋊D124C4, C6.58(D4×D5), D3023(C2×C4), Dic53(C4×S3), C10.60(S3×D4), C53(Dic35D4), C30.128(C2×D4), (C2×C20).197D6, D304C411C2, C30.53(C4○D4), C10.D414S3, (C2×C12).195D10, C32(Dic54D4), (C2×C30).96C23, C30.57(C22×C4), (Dic3×Dic5)⋊18C2, C2.4(D12⋊D5), C6.11(D42D5), C2.3(D10⋊D6), (C2×C60).167C22, (C2×Dic3).97D10, (C2×Dic5).105D6, (C22×S3).35D10, C10.31(Q83S3), (C6×Dic5).56C22, (C10×Dic3).56C22, (C2×Dic15).205C22, (C22×D15).104C22, C2.27(C4×S3×D5), C6.25(C2×C4×D5), (C2×C4×D15)⋊10C2, C10.58(S3×C2×C4), (C2×S3×Dic5)⋊2C2, (S3×C10)⋊9(C2×C4), (C5×D6⋊C4)⋊10C2, C22.51(C2×S3×D5), (C3×Dic5)⋊3(C2×C4), (C2×C4).180(S3×D5), (C2×C5⋊D12).6C2, (S3×C2×C10).13C22, (C3×C10.D4)⋊14C2, (C2×C6).108(C22×D5), (C2×C10).108(C22×S3), SmallGroup(480,482)

Series: Derived Chief Lower central Upper central

C1C30 — Dic1514D4
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — Dic1514D4
C15C30 — Dic1514D4
C1C22C2×C4

Generators and relations for Dic1514D4
 G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=a19, dad=a11, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1004 in 188 conjugacy classes, 60 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×3], C12 [×4], D6 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3, C5×S3 [×2], D15 [×2], C30 [×3], C4×D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, D6⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], C60, S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, Dic35D4, S3×Dic5 [×2], C5⋊D12 [×4], C6×Dic5 [×2], C10×Dic3, C4×D15 [×2], C2×Dic15, C2×C60, S3×C2×C10, C22×D15, Dic54D4, Dic3×Dic5, D304C4, C3×C10.D4, C5×D6⋊C4, C2×S3×Dic5, C2×C5⋊D12, C2×C4×D15, Dic1514D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4, Q83S3, S3×D5, C2×C4×D5, D4×D5, D42D5, Dic35D4, C2×S3×D5, Dic54D4, D12⋊D5, C4×S3×D5, D10⋊D6, Dic1514D4

Smallest permutation representation of Dic1514D4
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)
(1 119 16 104)(2 118 17 103)(3 117 18 102)(4 116 19 101)(5 115 20 100)(6 114 21 99)(7 113 22 98)(8 112 23 97)(9 111 24 96)(10 110 25 95)(11 109 26 94)(12 108 27 93)(13 107 28 92)(14 106 29 91)(15 105 30 120)(31 83 46 68)(32 82 47 67)(33 81 48 66)(34 80 49 65)(35 79 50 64)(36 78 51 63)(37 77 52 62)(38 76 53 61)(39 75 54 90)(40 74 55 89)(41 73 56 88)(42 72 57 87)(43 71 58 86)(44 70 59 85)(45 69 60 84)(121 213 136 228)(122 212 137 227)(123 211 138 226)(124 240 139 225)(125 239 140 224)(126 238 141 223)(127 237 142 222)(128 236 143 221)(129 235 144 220)(130 234 145 219)(131 233 146 218)(132 232 147 217)(133 231 148 216)(134 230 149 215)(135 229 150 214)(151 182 166 197)(152 181 167 196)(153 210 168 195)(154 209 169 194)(155 208 170 193)(156 207 171 192)(157 206 172 191)(158 205 173 190)(159 204 174 189)(160 203 175 188)(161 202 176 187)(162 201 177 186)(163 200 178 185)(164 199 179 184)(165 198 180 183)
(1 133 53 154)(2 122 54 173)(3 141 55 162)(4 130 56 151)(5 149 57 170)(6 138 58 159)(7 127 59 178)(8 146 60 167)(9 135 31 156)(10 124 32 175)(11 143 33 164)(12 132 34 153)(13 121 35 172)(14 140 36 161)(15 129 37 180)(16 148 38 169)(17 137 39 158)(18 126 40 177)(19 145 41 166)(20 134 42 155)(21 123 43 174)(22 142 44 163)(23 131 45 152)(24 150 46 171)(25 139 47 160)(26 128 48 179)(27 147 49 168)(28 136 50 157)(29 125 51 176)(30 144 52 165)(61 209 119 231)(62 198 120 220)(63 187 91 239)(64 206 92 228)(65 195 93 217)(66 184 94 236)(67 203 95 225)(68 192 96 214)(69 181 97 233)(70 200 98 222)(71 189 99 211)(72 208 100 230)(73 197 101 219)(74 186 102 238)(75 205 103 227)(76 194 104 216)(77 183 105 235)(78 202 106 224)(79 191 107 213)(80 210 108 232)(81 199 109 221)(82 188 110 240)(83 207 111 229)(84 196 112 218)(85 185 113 237)(86 204 114 226)(87 193 115 215)(88 182 116 234)(89 201 117 223)(90 190 118 212)
(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)(31 167)(32 178)(33 159)(34 170)(35 151)(36 162)(37 173)(38 154)(39 165)(40 176)(41 157)(42 168)(43 179)(44 160)(45 171)(46 152)(47 163)(48 174)(49 155)(50 166)(51 177)(52 158)(53 169)(54 180)(55 161)(56 172)(57 153)(58 164)(59 175)(60 156)(61 194)(62 205)(63 186)(64 197)(65 208)(66 189)(67 200)(68 181)(69 192)(70 203)(71 184)(72 195)(73 206)(74 187)(75 198)(76 209)(77 190)(78 201)(79 182)(80 193)(81 204)(82 185)(83 196)(84 207)(85 188)(86 199)(87 210)(88 191)(89 202)(90 183)(91 238)(92 219)(93 230)(94 211)(95 222)(96 233)(97 214)(98 225)(99 236)(100 217)(101 228)(102 239)(103 220)(104 231)(105 212)(106 223)(107 234)(108 215)(109 226)(110 237)(111 218)(112 229)(113 240)(114 221)(115 232)(116 213)(117 224)(118 235)(119 216)(120 227)

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), (1,119,16,104)(2,118,17,103)(3,117,18,102)(4,116,19,101)(5,115,20,100)(6,114,21,99)(7,113,22,98)(8,112,23,97)(9,111,24,96)(10,110,25,95)(11,109,26,94)(12,108,27,93)(13,107,28,92)(14,106,29,91)(15,105,30,120)(31,83,46,68)(32,82,47,67)(33,81,48,66)(34,80,49,65)(35,79,50,64)(36,78,51,63)(37,77,52,62)(38,76,53,61)(39,75,54,90)(40,74,55,89)(41,73,56,88)(42,72,57,87)(43,71,58,86)(44,70,59,85)(45,69,60,84)(121,213,136,228)(122,212,137,227)(123,211,138,226)(124,240,139,225)(125,239,140,224)(126,238,141,223)(127,237,142,222)(128,236,143,221)(129,235,144,220)(130,234,145,219)(131,233,146,218)(132,232,147,217)(133,231,148,216)(134,230,149,215)(135,229,150,214)(151,182,166,197)(152,181,167,196)(153,210,168,195)(154,209,169,194)(155,208,170,193)(156,207,171,192)(157,206,172,191)(158,205,173,190)(159,204,174,189)(160,203,175,188)(161,202,176,187)(162,201,177,186)(163,200,178,185)(164,199,179,184)(165,198,180,183), (1,133,53,154)(2,122,54,173)(3,141,55,162)(4,130,56,151)(5,149,57,170)(6,138,58,159)(7,127,59,178)(8,146,60,167)(9,135,31,156)(10,124,32,175)(11,143,33,164)(12,132,34,153)(13,121,35,172)(14,140,36,161)(15,129,37,180)(16,148,38,169)(17,137,39,158)(18,126,40,177)(19,145,41,166)(20,134,42,155)(21,123,43,174)(22,142,44,163)(23,131,45,152)(24,150,46,171)(25,139,47,160)(26,128,48,179)(27,147,49,168)(28,136,50,157)(29,125,51,176)(30,144,52,165)(61,209,119,231)(62,198,120,220)(63,187,91,239)(64,206,92,228)(65,195,93,217)(66,184,94,236)(67,203,95,225)(68,192,96,214)(69,181,97,233)(70,200,98,222)(71,189,99,211)(72,208,100,230)(73,197,101,219)(74,186,102,238)(75,205,103,227)(76,194,104,216)(77,183,105,235)(78,202,106,224)(79,191,107,213)(80,210,108,232)(81,199,109,221)(82,188,110,240)(83,207,111,229)(84,196,112,218)(85,185,113,237)(86,204,114,226)(87,193,115,215)(88,182,116,234)(89,201,117,223)(90,190,118,212), (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)(31,167)(32,178)(33,159)(34,170)(35,151)(36,162)(37,173)(38,154)(39,165)(40,176)(41,157)(42,168)(43,179)(44,160)(45,171)(46,152)(47,163)(48,174)(49,155)(50,166)(51,177)(52,158)(53,169)(54,180)(55,161)(56,172)(57,153)(58,164)(59,175)(60,156)(61,194)(62,205)(63,186)(64,197)(65,208)(66,189)(67,200)(68,181)(69,192)(70,203)(71,184)(72,195)(73,206)(74,187)(75,198)(76,209)(77,190)(78,201)(79,182)(80,193)(81,204)(82,185)(83,196)(84,207)(85,188)(86,199)(87,210)(88,191)(89,202)(90,183)(91,238)(92,219)(93,230)(94,211)(95,222)(96,233)(97,214)(98,225)(99,236)(100,217)(101,228)(102,239)(103,220)(104,231)(105,212)(106,223)(107,234)(108,215)(109,226)(110,237)(111,218)(112,229)(113,240)(114,221)(115,232)(116,213)(117,224)(118,235)(119,216)(120,227)>;

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), (1,119,16,104)(2,118,17,103)(3,117,18,102)(4,116,19,101)(5,115,20,100)(6,114,21,99)(7,113,22,98)(8,112,23,97)(9,111,24,96)(10,110,25,95)(11,109,26,94)(12,108,27,93)(13,107,28,92)(14,106,29,91)(15,105,30,120)(31,83,46,68)(32,82,47,67)(33,81,48,66)(34,80,49,65)(35,79,50,64)(36,78,51,63)(37,77,52,62)(38,76,53,61)(39,75,54,90)(40,74,55,89)(41,73,56,88)(42,72,57,87)(43,71,58,86)(44,70,59,85)(45,69,60,84)(121,213,136,228)(122,212,137,227)(123,211,138,226)(124,240,139,225)(125,239,140,224)(126,238,141,223)(127,237,142,222)(128,236,143,221)(129,235,144,220)(130,234,145,219)(131,233,146,218)(132,232,147,217)(133,231,148,216)(134,230,149,215)(135,229,150,214)(151,182,166,197)(152,181,167,196)(153,210,168,195)(154,209,169,194)(155,208,170,193)(156,207,171,192)(157,206,172,191)(158,205,173,190)(159,204,174,189)(160,203,175,188)(161,202,176,187)(162,201,177,186)(163,200,178,185)(164,199,179,184)(165,198,180,183), (1,133,53,154)(2,122,54,173)(3,141,55,162)(4,130,56,151)(5,149,57,170)(6,138,58,159)(7,127,59,178)(8,146,60,167)(9,135,31,156)(10,124,32,175)(11,143,33,164)(12,132,34,153)(13,121,35,172)(14,140,36,161)(15,129,37,180)(16,148,38,169)(17,137,39,158)(18,126,40,177)(19,145,41,166)(20,134,42,155)(21,123,43,174)(22,142,44,163)(23,131,45,152)(24,150,46,171)(25,139,47,160)(26,128,48,179)(27,147,49,168)(28,136,50,157)(29,125,51,176)(30,144,52,165)(61,209,119,231)(62,198,120,220)(63,187,91,239)(64,206,92,228)(65,195,93,217)(66,184,94,236)(67,203,95,225)(68,192,96,214)(69,181,97,233)(70,200,98,222)(71,189,99,211)(72,208,100,230)(73,197,101,219)(74,186,102,238)(75,205,103,227)(76,194,104,216)(77,183,105,235)(78,202,106,224)(79,191,107,213)(80,210,108,232)(81,199,109,221)(82,188,110,240)(83,207,111,229)(84,196,112,218)(85,185,113,237)(86,204,114,226)(87,193,115,215)(88,182,116,234)(89,201,117,223)(90,190,118,212), (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)(31,167)(32,178)(33,159)(34,170)(35,151)(36,162)(37,173)(38,154)(39,165)(40,176)(41,157)(42,168)(43,179)(44,160)(45,171)(46,152)(47,163)(48,174)(49,155)(50,166)(51,177)(52,158)(53,169)(54,180)(55,161)(56,172)(57,153)(58,164)(59,175)(60,156)(61,194)(62,205)(63,186)(64,197)(65,208)(66,189)(67,200)(68,181)(69,192)(70,203)(71,184)(72,195)(73,206)(74,187)(75,198)(76,209)(77,190)(78,201)(79,182)(80,193)(81,204)(82,185)(83,196)(84,207)(85,188)(86,199)(87,210)(88,191)(89,202)(90,183)(91,238)(92,219)(93,230)(94,211)(95,222)(96,233)(97,214)(98,225)(99,236)(100,217)(101,228)(102,239)(103,220)(104,231)(105,212)(106,223)(107,234)(108,215)(109,226)(110,237)(111,218)(112,229)(113,240)(114,221)(115,232)(116,213)(117,224)(118,235)(119,216)(120,227) );

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)], [(1,119,16,104),(2,118,17,103),(3,117,18,102),(4,116,19,101),(5,115,20,100),(6,114,21,99),(7,113,22,98),(8,112,23,97),(9,111,24,96),(10,110,25,95),(11,109,26,94),(12,108,27,93),(13,107,28,92),(14,106,29,91),(15,105,30,120),(31,83,46,68),(32,82,47,67),(33,81,48,66),(34,80,49,65),(35,79,50,64),(36,78,51,63),(37,77,52,62),(38,76,53,61),(39,75,54,90),(40,74,55,89),(41,73,56,88),(42,72,57,87),(43,71,58,86),(44,70,59,85),(45,69,60,84),(121,213,136,228),(122,212,137,227),(123,211,138,226),(124,240,139,225),(125,239,140,224),(126,238,141,223),(127,237,142,222),(128,236,143,221),(129,235,144,220),(130,234,145,219),(131,233,146,218),(132,232,147,217),(133,231,148,216),(134,230,149,215),(135,229,150,214),(151,182,166,197),(152,181,167,196),(153,210,168,195),(154,209,169,194),(155,208,170,193),(156,207,171,192),(157,206,172,191),(158,205,173,190),(159,204,174,189),(160,203,175,188),(161,202,176,187),(162,201,177,186),(163,200,178,185),(164,199,179,184),(165,198,180,183)], [(1,133,53,154),(2,122,54,173),(3,141,55,162),(4,130,56,151),(5,149,57,170),(6,138,58,159),(7,127,59,178),(8,146,60,167),(9,135,31,156),(10,124,32,175),(11,143,33,164),(12,132,34,153),(13,121,35,172),(14,140,36,161),(15,129,37,180),(16,148,38,169),(17,137,39,158),(18,126,40,177),(19,145,41,166),(20,134,42,155),(21,123,43,174),(22,142,44,163),(23,131,45,152),(24,150,46,171),(25,139,47,160),(26,128,48,179),(27,147,49,168),(28,136,50,157),(29,125,51,176),(30,144,52,165),(61,209,119,231),(62,198,120,220),(63,187,91,239),(64,206,92,228),(65,195,93,217),(66,184,94,236),(67,203,95,225),(68,192,96,214),(69,181,97,233),(70,200,98,222),(71,189,99,211),(72,208,100,230),(73,197,101,219),(74,186,102,238),(75,205,103,227),(76,194,104,216),(77,183,105,235),(78,202,106,224),(79,191,107,213),(80,210,108,232),(81,199,109,221),(82,188,110,240),(83,207,111,229),(84,196,112,218),(85,185,113,237),(86,204,114,226),(87,193,115,215),(88,182,116,234),(89,201,117,223),(90,190,118,212)], [(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),(31,167),(32,178),(33,159),(34,170),(35,151),(36,162),(37,173),(38,154),(39,165),(40,176),(41,157),(42,168),(43,179),(44,160),(45,171),(46,152),(47,163),(48,174),(49,155),(50,166),(51,177),(52,158),(53,169),(54,180),(55,161),(56,172),(57,153),(58,164),(59,175),(60,156),(61,194),(62,205),(63,186),(64,197),(65,208),(66,189),(67,200),(68,181),(69,192),(70,203),(71,184),(72,195),(73,206),(74,187),(75,198),(76,209),(77,190),(78,201),(79,182),(80,193),(81,204),(82,185),(83,196),(84,207),(85,188),(86,199),(87,210),(88,191),(89,202),(90,183),(91,238),(92,219),(93,230),(94,211),(95,222),(96,233),(97,214),(98,225),(99,236),(100,217),(101,228),(102,239),(103,220),(104,231),(105,212),(106,223),(107,234),(108,215),(109,226),(110,237),(111,218),(112,229),(113,240),(114,221),(115,232),(116,213),(117,224),(118,235),(119,216),(120,227)])

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222222234444444444445566610···10101010101212121212121515202020202020202030···3060···60
size1111663030222661010101015151515222222···2121212124420202020444444121212124···44···4

66 irreducible representations

dim11111111122222222222444444444
type++++++++++++++++++++-++
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6C4○D4D10D10D10C4×S3C4×D5S3×D4Q83S3S3×D5D4×D5D42D5C2×S3×D5D12⋊D5C4×S3×D5D10⋊D6
kernelDic1514D4Dic3×Dic5D304C4C3×C10.D4C5×D6⋊C4C2×S3×Dic5C2×C5⋊D12C2×C4×D15C5⋊D12C10.D4Dic15D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3Dic5D6C10C10C2×C4C6C6C22C2C2C2
# reps11111111812221222248112222444

Matrix representation of Dic1514D4 in GL6(𝔽61)

1600000
19430000
0024100
00526000
000010
000001
,
15500000
15460000
00503700
0001100
000010
000001
,
18600000
18430000
001000
000100
0000858
00004253
,
100000
010000
0014100
0006000
0000533
0000408

G:=sub<GL(6,GF(61))| [1,19,0,0,0,0,60,43,0,0,0,0,0,0,2,52,0,0,0,0,41,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,15,0,0,0,0,50,46,0,0,0,0,0,0,50,0,0,0,0,0,37,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,18,0,0,0,0,60,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,42,0,0,0,0,58,53],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,41,60,0,0,0,0,0,0,53,40,0,0,0,0,3,8] >;

Dic1514D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_{14}D_4
% in TeX

G:=Group("Dic15:14D4");
// GroupNames label

G:=SmallGroup(480,482);
// by ID

G=gap.SmallGroup(480,482);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,135,100,1356,18822]);
// 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^19,d*a*d=a^11,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽