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G = Dic1516D4order 480 = 25·3·5

6th semidirect product of Dic15 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1516D4, C1529(C4×D4), C54(D4×Dic3), C6.87(D4×D5), C10.88(S3×D4), C5⋊D42Dic3, D105(C2×Dic3), C30.239(C2×D4), C223(D5×Dic3), C23.51(S3×D5), Dic53(C2×Dic3), C6.D410D5, C37(Dic54D4), C30.Q835C2, (C22×D5).65D6, (C22×C10).53D6, (C22×C6).38D10, (Dic3×Dic5)⋊37C2, C30.153(C4○D4), C2.5(D10⋊D6), D10⋊Dic332C2, C6.83(D42D5), (C2×C30).201C23, C30.149(C22×C4), (C2×Dic5).131D6, C10.83(D42S3), (C2×Dic3).124D10, C2.6(C30.C23), (C22×Dic15)⋊14C2, (C22×C30).63C22, C10.33(C22×Dic3), (C6×Dic5).117C22, (C2×Dic15).227C22, (C10×Dic3).117C22, (C2×C6)⋊3(C4×D5), C6.96(C2×C4×D5), (C3×C5⋊D4)⋊4C4, (C2×C30)⋊20(C2×C4), (C6×D5)⋊10(C2×C4), (C2×D5×Dic3)⋊18C2, (C6×C5⋊D4).7C2, (C2×C5⋊D4).6S3, C2.20(C2×D5×Dic3), C22.89(C2×S3×D5), (C3×Dic5)⋊6(C2×C4), (C2×C10)⋊9(C2×Dic3), (D5×C2×C6).52C22, (C5×C6.D4)⋊12C2, (C2×C6).213(C22×D5), (C2×C10).213(C22×S3), SmallGroup(480,635)

Series: Derived Chief Lower central Upper central

C1C30 — Dic1516D4
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — Dic1516D4
C15C30 — Dic1516D4
C1C22C23

Generators and relations for Dic1516D4
 G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a19, bc=cb, bd=db, dcd=c-1 >

Subgroups: 828 in 188 conjugacy classes, 70 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×9], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12, C3×D4 [×4], C22×C6, C22×C6, C3×D5 [×2], C30 [×3], C30 [×2], C4×D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×C10, C4×Dic3, C4⋊Dic3, C6.D4, C6.D4, C22×Dic3 [×2], C6×D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], Dic15, C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, D4×Dic3, D5×Dic3 [×2], C6×Dic5, C3×C5⋊D4 [×4], C10×Dic3 [×2], C2×Dic15 [×2], C2×Dic15 [×2], D5×C2×C6, C22×C30, Dic54D4, Dic3×Dic5, D10⋊Dic3, C30.Q8, C5×C6.D4, C2×D5×Dic3, C6×C5⋊D4, C22×Dic15, Dic1516D4
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, D42S3, C22×Dic3, S3×D5, C2×C4×D5, D4×D5, D42D5, D4×Dic3, D5×Dic3 [×2], C2×S3×D5, Dic54D4, C2×D5×Dic3, C30.C23, D10⋊D6, Dic1516D4

Smallest permutation representation of Dic1516D4
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 32 16 47)(2 31 17 46)(3 60 18 45)(4 59 19 44)(5 58 20 43)(6 57 21 42)(7 56 22 41)(8 55 23 40)(9 54 24 39)(10 53 25 38)(11 52 26 37)(12 51 27 36)(13 50 28 35)(14 49 29 34)(15 48 30 33)(61 173 76 158)(62 172 77 157)(63 171 78 156)(64 170 79 155)(65 169 80 154)(66 168 81 153)(67 167 82 152)(68 166 83 151)(69 165 84 180)(70 164 85 179)(71 163 86 178)(72 162 87 177)(73 161 88 176)(74 160 89 175)(75 159 90 174)(91 123 106 138)(92 122 107 137)(93 121 108 136)(94 150 109 135)(95 149 110 134)(96 148 111 133)(97 147 112 132)(98 146 113 131)(99 145 114 130)(100 144 115 129)(101 143 116 128)(102 142 117 127)(103 141 118 126)(104 140 119 125)(105 139 120 124)(181 230 196 215)(182 229 197 214)(183 228 198 213)(184 227 199 212)(185 226 200 211)(186 225 201 240)(187 224 202 239)(188 223 203 238)(189 222 204 237)(190 221 205 236)(191 220 206 235)(192 219 207 234)(193 218 208 233)(194 217 209 232)(195 216 210 231)
(1 100 168 196)(2 119 169 185)(3 108 170 204)(4 97 171 193)(5 116 172 182)(6 105 173 201)(7 94 174 190)(8 113 175 209)(9 102 176 198)(10 91 177 187)(11 110 178 206)(12 99 179 195)(13 118 180 184)(14 107 151 203)(15 96 152 192)(16 115 153 181)(17 104 154 200)(18 93 155 189)(19 112 156 208)(20 101 157 197)(21 120 158 186)(22 109 159 205)(23 98 160 194)(24 117 161 183)(25 106 162 202)(26 95 163 191)(27 114 164 210)(28 103 165 199)(29 92 166 188)(30 111 167 207)(31 125 80 226)(32 144 81 215)(33 133 82 234)(34 122 83 223)(35 141 84 212)(36 130 85 231)(37 149 86 220)(38 138 87 239)(39 127 88 228)(40 146 89 217)(41 135 90 236)(42 124 61 225)(43 143 62 214)(44 132 63 233)(45 121 64 222)(46 140 65 211)(47 129 66 230)(48 148 67 219)(49 137 68 238)(50 126 69 227)(51 145 70 216)(52 134 71 235)(53 123 72 224)(54 142 73 213)(55 131 74 232)(56 150 75 221)(57 139 76 240)(58 128 77 229)(59 147 78 218)(60 136 79 237)
(1 168)(2 157)(3 176)(4 165)(5 154)(6 173)(7 162)(8 151)(9 170)(10 159)(11 178)(12 167)(13 156)(14 175)(15 164)(16 153)(17 172)(18 161)(19 180)(20 169)(21 158)(22 177)(23 166)(24 155)(25 174)(26 163)(27 152)(28 171)(29 160)(30 179)(31 62)(32 81)(33 70)(34 89)(35 78)(36 67)(37 86)(38 75)(39 64)(40 83)(41 72)(42 61)(43 80)(44 69)(45 88)(46 77)(47 66)(48 85)(49 74)(50 63)(51 82)(52 71)(53 90)(54 79)(55 68)(56 87)(57 76)(58 65)(59 84)(60 73)(91 109)(92 98)(93 117)(94 106)(96 114)(97 103)(99 111)(101 119)(102 108)(104 116)(107 113)(112 118)(121 127)(122 146)(123 135)(125 143)(126 132)(128 140)(130 148)(131 137)(133 145)(136 142)(138 150)(141 147)(182 200)(183 189)(184 208)(185 197)(187 205)(188 194)(190 202)(192 210)(193 199)(195 207)(198 204)(203 209)(211 229)(212 218)(213 237)(214 226)(216 234)(217 223)(219 231)(221 239)(222 228)(224 236)(227 233)(232 238)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J12A12B15A15B20A···20H30A···30N
order12222222344444444444455666666610···10101010101212151520···2030···30
size1111221010266661010151515153030222224420202···2444420204412···124···4

66 irreducible representations

dim11111111122222222222444444444
type++++++++++++-+++++-++--+-+
imageC1C2C2C2C2C2C2C2C4S3D4D5D6Dic3D6D6C4○D4D10D10C4×D5S3×D4D42S3S3×D5D4×D5D42D5D5×Dic3C2×S3×D5C30.C23D10⋊D6
kernelDic1516D4Dic3×Dic5D10⋊Dic3C30.Q8C5×C6.D4C2×D5×Dic3C6×C5⋊D4C22×Dic15C3×C5⋊D4C2×C5⋊D4Dic15C6.D4C2×Dic5C5⋊D4C22×D5C22×C10C30C2×Dic3C22×C6C2×C6C10C10C23C6C6C22C22C2C2
# reps11111111812214112428112224244

Matrix representation of Dic1516D4 in GL6(𝔽61)

6000000
0600000
0006000
0011800
0000160
000010
,
1100000
0110000
0011000
00465000
00003651
00002625
,
0300000
200000
0060000
0018100
0000600
0000060
,
6000000
010000
001000
00436000
000010
000001

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,60,18,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,46,0,0,0,0,0,50,0,0,0,0,0,0,36,26,0,0,0,0,51,25],[0,2,0,0,0,0,30,0,0,0,0,0,0,0,60,18,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,43,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic1516D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_{16}D_4
% in TeX

G:=Group("Dic15:16D4");
// GroupNames label

G:=SmallGroup(480,635);
// by ID

G=gap.SmallGroup(480,635);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,219,1356,18822]);
// 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^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽