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G = Dic15.D4order 480 = 25·3·5

6th non-split extension by Dic15 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D103Dic6, Dic15.6D4, (C6×D5)⋊3Q8, C4⋊Dic38D5, C6.39(D4×D5), C6.35(Q8×D5), C30.51(C2×D4), C10.40(S3×D4), C30.47(C2×Q8), C33(D10⋊Q8), (C2×C12).22D10, (C2×C20).229D6, C1518(C22⋊Q8), C2.17(D5×Dic6), D10⋊C4.9S3, C6.73(C4○D20), Dic155C419C2, C30.4Q820C2, (C2×Dic5).38D6, C10.17(C2×Dic6), (C22×D5).52D6, C30.122(C4○D4), C2.18(C20⋊D6), (C2×C30).120C23, (C2×C60).322C22, (C2×Dic3).38D10, C53(Dic3.D4), C10.47(D42S3), D10⋊Dic3.13C2, (C6×Dic5).73C22, C2.19(Dic5.D6), (C2×Dic15).97C22, (C10×Dic3).74C22, (C2×C15⋊Q8)⋊10C2, (C2×C4).53(S3×D5), (C2×D5×Dic3).7C2, (C5×C4⋊Dic3)⋊19C2, (D5×C2×C6).22C22, C22.183(C2×S3×D5), (C2×C6).132(C22×D5), (C3×D10⋊C4).12C2, (C2×C10).132(C22×S3), SmallGroup(480,506)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic15.D4
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — Dic15.D4
C15C2×C30 — Dic15.D4
C1C22C2×C4

Generators and relations for Dic15.D4
 G = < a,b,c,d | a30=c4=1, b2=d2=a15, bab-1=a-1, ac=ca, dad-1=a11, cbc-1=dbd-1=a15b, dcd-1=c-1 >

Subgroups: 748 in 148 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20 [×3], D10 [×2], D10 [×2], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12, C2×C12, C22×C6, C3×D5 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, C6×D5 [×2], C6×D5 [×2], C2×C30, C10.D4 [×2], D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, Dic3.D4, D5×Dic3 [×2], C15⋊Q8 [×2], C6×Dic5, C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, D10⋊Q8, D10⋊Dic3, Dic155C4, C3×D10⋊C4, C5×C4⋊Dic3, C30.4Q8, C2×D5×Dic3, C2×C15⋊Q8, Dic15.D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], Dic6 [×2], C22×S3, C22⋊Q8, C22×D5, C2×Dic6, S3×D4, D42S3, S3×D5, C4○D20, D4×D5, Q8×D5, Dic3.D4, C2×S3×D5, D10⋊Q8, D5×Dic6, C20⋊D6, Dic5.D6, Dic15.D4

Smallest permutation representation of Dic15.D4
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 70 16 85)(2 69 17 84)(3 68 18 83)(4 67 19 82)(5 66 20 81)(6 65 21 80)(7 64 22 79)(8 63 23 78)(9 62 24 77)(10 61 25 76)(11 90 26 75)(12 89 27 74)(13 88 28 73)(14 87 29 72)(15 86 30 71)(31 174 46 159)(32 173 47 158)(33 172 48 157)(34 171 49 156)(35 170 50 155)(36 169 51 154)(37 168 52 153)(38 167 53 152)(39 166 54 151)(40 165 55 180)(41 164 56 179)(42 163 57 178)(43 162 58 177)(44 161 59 176)(45 160 60 175)(91 239 106 224)(92 238 107 223)(93 237 108 222)(94 236 109 221)(95 235 110 220)(96 234 111 219)(97 233 112 218)(98 232 113 217)(99 231 114 216)(100 230 115 215)(101 229 116 214)(102 228 117 213)(103 227 118 212)(104 226 119 211)(105 225 120 240)(121 195 136 210)(122 194 137 209)(123 193 138 208)(124 192 139 207)(125 191 140 206)(126 190 141 205)(127 189 142 204)(128 188 143 203)(129 187 144 202)(130 186 145 201)(131 185 146 200)(132 184 147 199)(133 183 148 198)(134 182 149 197)(135 181 150 196)
(1 138 172 227)(2 139 173 228)(3 140 174 229)(4 141 175 230)(5 142 176 231)(6 143 177 232)(7 144 178 233)(8 145 179 234)(9 146 180 235)(10 147 151 236)(11 148 152 237)(12 149 153 238)(13 150 154 239)(14 121 155 240)(15 122 156 211)(16 123 157 212)(17 124 158 213)(18 125 159 214)(19 126 160 215)(20 127 161 216)(21 128 162 217)(22 129 163 218)(23 130 164 219)(24 131 165 220)(25 132 166 221)(26 133 167 222)(27 134 168 223)(28 135 169 224)(29 136 170 225)(30 137 171 226)(31 116 83 206)(32 117 84 207)(33 118 85 208)(34 119 86 209)(35 120 87 210)(36 91 88 181)(37 92 89 182)(38 93 90 183)(39 94 61 184)(40 95 62 185)(41 96 63 186)(42 97 64 187)(43 98 65 188)(44 99 66 189)(45 100 67 190)(46 101 68 191)(47 102 69 192)(48 103 70 193)(49 104 71 194)(50 105 72 195)(51 106 73 196)(52 107 74 197)(53 108 75 198)(54 109 76 199)(55 110 77 200)(56 111 78 201)(57 112 79 202)(58 113 80 203)(59 114 81 204)(60 115 82 205)
(1 193 16 208)(2 204 17 189)(3 185 18 200)(4 196 19 181)(5 207 20 192)(6 188 21 203)(7 199 22 184)(8 210 23 195)(9 191 24 206)(10 202 25 187)(11 183 26 198)(12 194 27 209)(13 205 28 190)(14 186 29 201)(15 197 30 182)(31 235 46 220)(32 216 47 231)(33 227 48 212)(34 238 49 223)(35 219 50 234)(36 230 51 215)(37 211 52 226)(38 222 53 237)(39 233 54 218)(40 214 55 229)(41 225 56 240)(42 236 57 221)(43 217 58 232)(44 228 59 213)(45 239 60 224)(61 144 76 129)(62 125 77 140)(63 136 78 121)(64 147 79 132)(65 128 80 143)(66 139 81 124)(67 150 82 135)(68 131 83 146)(69 142 84 127)(70 123 85 138)(71 134 86 149)(72 145 87 130)(73 126 88 141)(74 137 89 122)(75 148 90 133)(91 175 106 160)(92 156 107 171)(93 167 108 152)(94 178 109 163)(95 159 110 174)(96 170 111 155)(97 151 112 166)(98 162 113 177)(99 173 114 158)(100 154 115 169)(101 165 116 180)(102 176 117 161)(103 157 118 172)(104 168 119 153)(105 179 120 164)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L30A···30F60A···60H
order122222344444444556666610···101212121215152020202020···2030···3060···60
size11111010246612203030602222220202···244202044444412···124···44···4

60 irreducible representations

dim11111111222222222222444444444
type++++++++++-++++++-+-++-+-
imageC1C2C2C2C2C2C2C2S3D4Q8D5D6D6D6C4○D4D10D10Dic6C4○D20S3×D4D42S3S3×D5D4×D5Q8×D5C2×S3×D5D5×Dic6C20⋊D6Dic5.D6
kernelDic15.D4D10⋊Dic3Dic155C4C3×D10⋊C4C5×C4⋊Dic3C30.4Q8C2×D5×Dic3C2×C15⋊Q8D10⋊C4Dic15C6×D5C4⋊Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12D10C6C10C10C2×C4C6C6C22C2C2C2
# reps11111111122211124248112222444

Matrix representation of Dic15.D4 in GL6(𝔽61)

010000
60430000
001000
000100
0000215
00001260
,
6000000
1810000
001000
000100
0000573
0000354
,
6000000
0600000
0052500
008900
00005342
0000588
,
100000
010000
0095600
00165200
00003822
00004823

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,12,0,0,0,0,15,60],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,57,35,0,0,0,0,3,4],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,52,8,0,0,0,0,5,9,0,0,0,0,0,0,53,58,0,0,0,0,42,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,56,52,0,0,0,0,0,0,38,48,0,0,0,0,22,23] >;

Dic15.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}.D_4
% in TeX

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

G:=SmallGroup(480,506);
// by ID

G=gap.SmallGroup(480,506);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=1,b^2=d^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽