Copied to
clipboard

G = Dic159D4order 480 = 25·3·5

4th semidirect product of Dic15 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic159D4, D64(C4×D5), C1518(C4×D4), D6⋊C411D5, D106(C4×S3), C15⋊D45C4, C6.45(D4×D5), C30.57(C2×D4), C10.46(S3×D4), (C2×C20).199D6, D10⋊C411S3, Dic1518(C2×C4), (C4×Dic15)⋊12C2, (C2×C12).197D10, C54(Dic34D4), C33(Dic54D4), C2.4(C20⋊D6), C30.61(C22×C4), Dic155C425C2, (C22×D5).56D6, C30.125(C4○D4), C6.73(D42D5), (C2×C30).132C23, (C2×C60).168C22, (C2×Dic5).116D6, (C22×S3).46D10, C10.72(D42S3), (C2×Dic3).111D10, C2.4(C30.C23), (C6×Dic5).82C22, (C10×Dic3).83C22, (C2×Dic15).210C22, C2.31(C4×S3×D5), C6.29(C2×C4×D5), (C6×D5)⋊6(C2×C4), C10.62(S3×C2×C4), (C2×S3×Dic5)⋊9C2, (C5×D6⋊C4)⋊11C2, (C2×D5×Dic3)⋊11C2, (S3×C10)⋊13(C2×C4), C22.64(C2×S3×D5), (C2×C4).181(S3×D5), (C2×C15⋊D4).4C2, (D5×C2×C6).26C22, (S3×C2×C10).28C22, (C3×D10⋊C4)⋊11C2, (C2×C6).144(C22×D5), (C2×C10).144(C22×S3), SmallGroup(480,518)

Series: Derived Chief Lower central Upper central

C1C30 — Dic159D4
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — Dic159D4
C15C30 — Dic159D4
C1C22C2×C4

Generators and relations for Dic159D4
 G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=cac-1=a-1, dad=a19, bc=cb, bd=db, dcd=c-1 >

Subgroups: 908 in 188 conjugacy classes, 60 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×8], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×5], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×5], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C2×C12, C22×S3, C22×C6, C5×S3 [×2], C3×D5 [×2], C30 [×3], C4×D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15 [×4], C60, C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, Dic34D4, D5×Dic3 [×2], S3×Dic5 [×2], C15⋊D4 [×4], C6×Dic5, C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, Dic54D4, Dic155C4, C3×D10⋊C4, C5×D6⋊C4, C4×Dic15, C2×D5×Dic3, C2×S3×Dic5, C2×C15⋊D4, Dic159D4
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, D42S3, S3×D5, C2×C4×D5, D4×D5, D42D5, Dic34D4, C2×S3×D5, Dic54D4, C4×S3×D5, C20⋊D6, C30.C23, Dic159D4

Smallest permutation representation of Dic159D4
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 39 16 54)(2 38 17 53)(3 37 18 52)(4 36 19 51)(5 35 20 50)(6 34 21 49)(7 33 22 48)(8 32 23 47)(9 31 24 46)(10 60 25 45)(11 59 26 44)(12 58 27 43)(13 57 28 42)(14 56 29 41)(15 55 30 40)(61 227 76 212)(62 226 77 211)(63 225 78 240)(64 224 79 239)(65 223 80 238)(66 222 81 237)(67 221 82 236)(68 220 83 235)(69 219 84 234)(70 218 85 233)(71 217 86 232)(72 216 87 231)(73 215 88 230)(74 214 89 229)(75 213 90 228)(91 202 106 187)(92 201 107 186)(93 200 108 185)(94 199 109 184)(95 198 110 183)(96 197 111 182)(97 196 112 181)(98 195 113 210)(99 194 114 209)(100 193 115 208)(101 192 116 207)(102 191 117 206)(103 190 118 205)(104 189 119 204)(105 188 120 203)(121 171 136 156)(122 170 137 155)(123 169 138 154)(124 168 139 153)(125 167 140 152)(126 166 141 151)(127 165 142 180)(128 164 143 179)(129 163 144 178)(130 162 145 177)(131 161 146 176)(132 160 147 175)(133 159 148 174)(134 158 149 173)(135 157 150 172)
(1 206 78 142)(2 205 79 141)(3 204 80 140)(4 203 81 139)(5 202 82 138)(6 201 83 137)(7 200 84 136)(8 199 85 135)(9 198 86 134)(10 197 87 133)(11 196 88 132)(12 195 89 131)(13 194 90 130)(14 193 61 129)(15 192 62 128)(16 191 63 127)(17 190 64 126)(18 189 65 125)(19 188 66 124)(20 187 67 123)(21 186 68 122)(22 185 69 121)(23 184 70 150)(24 183 71 149)(25 182 72 148)(26 181 73 147)(27 210 74 146)(28 209 75 145)(29 208 76 144)(30 207 77 143)(31 110 232 158)(32 109 233 157)(33 108 234 156)(34 107 235 155)(35 106 236 154)(36 105 237 153)(37 104 238 152)(38 103 239 151)(39 102 240 180)(40 101 211 179)(41 100 212 178)(42 99 213 177)(43 98 214 176)(44 97 215 175)(45 96 216 174)(46 95 217 173)(47 94 218 172)(48 93 219 171)(49 92 220 170)(50 91 221 169)(51 120 222 168)(52 119 223 167)(53 118 224 166)(54 117 225 165)(55 116 226 164)(56 115 227 163)(57 114 228 162)(58 113 229 161)(59 112 230 160)(60 111 231 159)
(1 78)(2 67)(3 86)(4 75)(5 64)(6 83)(7 72)(8 61)(9 80)(10 69)(11 88)(12 77)(13 66)(14 85)(15 74)(16 63)(17 82)(18 71)(19 90)(20 79)(21 68)(22 87)(23 76)(24 65)(25 84)(26 73)(27 62)(28 81)(29 70)(30 89)(31 238)(32 227)(33 216)(34 235)(35 224)(36 213)(37 232)(38 221)(39 240)(40 229)(41 218)(42 237)(43 226)(44 215)(45 234)(46 223)(47 212)(48 231)(49 220)(50 239)(51 228)(52 217)(53 236)(54 225)(55 214)(56 233)(57 222)(58 211)(59 230)(60 219)(91 103)(93 111)(94 100)(95 119)(96 108)(98 116)(99 105)(101 113)(104 110)(106 118)(109 115)(114 120)(121 133)(123 141)(124 130)(125 149)(126 138)(128 146)(129 135)(131 143)(134 140)(136 148)(139 145)(144 150)(151 169)(152 158)(153 177)(154 166)(156 174)(157 163)(159 171)(161 179)(162 168)(164 176)(167 173)(172 178)(182 200)(183 189)(184 208)(185 197)(187 205)(188 194)(190 202)(192 210)(193 199)(195 207)(198 204)(203 209)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order122222223444444444444556666610···1010101010121212121515202020202020202030···3060···60
size11116610102226610101515151530302222220202···212121212442020444444121212124···44···4

66 irreducible representations

dim111111111222222222222444444444
type++++++++++++++++++-++-+-
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6D6C4○D4D10D10D10C4×S3C4×D5S3×D4D42S3S3×D5D4×D5D42D5C2×S3×D5C4×S3×D5C20⋊D6C30.C23
kernelDic159D4Dic155C4C3×D10⋊C4C5×D6⋊C4C4×Dic15C2×D5×Dic3C2×S3×Dic5C2×C15⋊D4C15⋊D4D10⋊C4Dic15D6⋊C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C22×S3D10D6C10C10C2×C4C6C6C22C2C2C2
# reps111111118122111222248112222444

Matrix representation of Dic159D4 in GL6(𝔽61)

100000
010000
0060100
00421800
0000160
000010
,
100000
010000
0043100
00431800
0000430
00003457
,
4020000
23210000
0043100
00431800
00004425
0000817
,
100000
21600000
0043100
00431800
000010
000001

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,42,0,0,0,0,1,18,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,4,34,0,0,0,0,30,57],[40,23,0,0,0,0,2,21,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,44,8,0,0,0,0,25,17],[1,21,0,0,0,0,0,60,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic159D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_9D_4
% in TeX

G:=Group("Dic15:9D4");
// GroupNames label

G:=SmallGroup(480,518);
// by ID

G=gap.SmallGroup(480,518);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽