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## G = Dic15⋊D4order 480 = 25·3·5

### 1st semidirect product of Dic15 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15⋊D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — Dic15⋊D4
 Lower central C15 — C2×C30 — Dic15⋊D4
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic15⋊D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, ac=ca, dad=a19, cbc-1=dbd=a15b, dcd=c-1 >

Subgroups: 1052 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3, C6 [×3], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×3], C10 [×3], C10, Dic3 [×4], C12, D6 [×3], C2×C6, C2×C6 [×7], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C2×Dic3, C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C5×S3, C3×D5 [×3], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20, C22×D5 [×2], C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C5×Dic3, Dic15 [×2], Dic15, C60, C6×D5 [×2], C6×D5 [×5], S3×C10 [×3], C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4 [×2], C23.14D6, D5×Dic3 [×2], C15⋊D4 [×4], C3×D20 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], S3×C2×C10, D10⋊D4, D10⋊Dic3, C5×D6⋊C4, C30.4Q8, C2×D5×Dic3, C2×C15⋊D4 [×2], C6×D20, Dic15⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊D4, C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5 [×2], C23.14D6, C2×S3×D5, D10⋊D4, D205S3, C20⋊D6, D5×C3⋊D4, 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)
(1 166 16 151)(2 165 17 180)(3 164 18 179)(4 163 19 178)(5 162 20 177)(6 161 21 176)(7 160 22 175)(8 159 23 174)(9 158 24 173)(10 157 25 172)(11 156 26 171)(12 155 27 170)(13 154 28 169)(14 153 29 168)(15 152 30 167)(31 205 46 190)(32 204 47 189)(33 203 48 188)(34 202 49 187)(35 201 50 186)(36 200 51 185)(37 199 52 184)(38 198 53 183)(39 197 54 182)(40 196 55 181)(41 195 56 210)(42 194 57 209)(43 193 58 208)(44 192 59 207)(45 191 60 206)(61 211 76 226)(62 240 77 225)(63 239 78 224)(64 238 79 223)(65 237 80 222)(66 236 81 221)(67 235 82 220)(68 234 83 219)(69 233 84 218)(70 232 85 217)(71 231 86 216)(72 230 87 215)(73 229 88 214)(74 228 89 213)(75 227 90 212)(91 144 106 129)(92 143 107 128)(93 142 108 127)(94 141 109 126)(95 140 110 125)(96 139 111 124)(97 138 112 123)(98 137 113 122)(99 136 114 121)(100 135 115 150)(101 134 116 149)(102 133 117 148)(103 132 118 147)(104 131 119 146)(105 130 120 145)
(1 231 115 60)(2 232 116 31)(3 233 117 32)(4 234 118 33)(5 235 119 34)(6 236 120 35)(7 237 91 36)(8 238 92 37)(9 239 93 38)(10 240 94 39)(11 211 95 40)(12 212 96 41)(13 213 97 42)(14 214 98 43)(15 215 99 44)(16 216 100 45)(17 217 101 46)(18 218 102 47)(19 219 103 48)(20 220 104 49)(21 221 105 50)(22 222 106 51)(23 223 107 52)(24 224 108 53)(25 225 109 54)(26 226 110 55)(27 227 111 56)(28 228 112 57)(29 229 113 58)(30 230 114 59)(61 140 181 156)(62 141 182 157)(63 142 183 158)(64 143 184 159)(65 144 185 160)(66 145 186 161)(67 146 187 162)(68 147 188 163)(69 148 189 164)(70 149 190 165)(71 150 191 166)(72 121 192 167)(73 122 193 168)(74 123 194 169)(75 124 195 170)(76 125 196 171)(77 126 197 172)(78 127 198 173)(79 128 199 174)(80 129 200 175)(81 130 201 176)(82 131 202 177)(83 132 203 178)(84 133 204 179)(85 134 205 180)(86 135 206 151)(87 136 207 152)(88 137 208 153)(89 138 209 154)(90 139 210 155)
(1 216)(2 235)(3 224)(4 213)(5 232)(6 221)(7 240)(8 229)(9 218)(10 237)(11 226)(12 215)(13 234)(14 223)(15 212)(16 231)(17 220)(18 239)(19 228)(20 217)(21 236)(22 225)(23 214)(24 233)(25 222)(26 211)(27 230)(28 219)(29 238)(30 227)(31 119)(32 108)(33 97)(34 116)(35 105)(36 94)(37 113)(38 102)(39 91)(40 110)(41 99)(42 118)(43 107)(44 96)(45 115)(46 104)(47 93)(48 112)(49 101)(50 120)(51 109)(52 98)(53 117)(54 106)(55 95)(56 114)(57 103)(58 92)(59 111)(60 100)(61 171)(62 160)(63 179)(64 168)(65 157)(66 176)(67 165)(68 154)(69 173)(70 162)(71 151)(72 170)(73 159)(74 178)(75 167)(76 156)(77 175)(78 164)(79 153)(80 172)(81 161)(82 180)(83 169)(84 158)(85 177)(86 166)(87 155)(88 174)(89 163)(90 152)(121 195)(122 184)(123 203)(124 192)(125 181)(126 200)(127 189)(128 208)(129 197)(130 186)(131 205)(132 194)(133 183)(134 202)(135 191)(136 210)(137 199)(138 188)(139 207)(140 196)(141 185)(142 204)(143 193)(144 182)(145 201)(146 190)(147 209)(148 198)(149 187)(150 206)

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

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

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

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 12A 12B 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 30A ··· 30F 60A ··· 60H 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 12 12 15 15 20 20 20 20 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 12 20 2 4 6 6 30 30 60 2 2 2 2 2 20 20 20 20 2 ··· 2 12 12 12 12 4 4 4 4 4 4 4 4 12 12 12 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 C4○D4 D10 D10 D10 C3⋊D4 C4○D20 S3×D4 D4⋊2S3 S3×D5 D4×D5 C2×S3×D5 D20⋊5S3 C20⋊D6 D5×C3⋊D4 kernel Dic15⋊D4 D10⋊Dic3 C5×D6⋊C4 C30.4Q8 C2×D5×Dic3 C2×C15⋊D4 C6×D20 C2×D20 Dic15 C6×D5 D6⋊C4 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C22×S3 D10 C6 C10 C10 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 2 4 8 1 1 2 4 2 4 4 4

Matrix representation of Dic15⋊D4 in GL4(𝔽61) generated by

 0 60 0 0 1 18 0 0 0 0 2 15 0 0 12 60
,
 11 0 0 0 46 50 0 0 0 0 46 52 0 0 59 15
,
 25 57 0 0 4 36 0 0 0 0 27 26 0 0 33 34
,
 25 57 0 0 34 36 0 0 0 0 34 35 0 0 28 27
G:=sub<GL(4,GF(61))| [0,1,0,0,60,18,0,0,0,0,2,12,0,0,15,60],[11,46,0,0,0,50,0,0,0,0,46,59,0,0,52,15],[25,4,0,0,57,36,0,0,0,0,27,33,0,0,26,34],[25,34,0,0,57,36,0,0,0,0,34,28,0,0,35,27] >;

Dic15⋊D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,484);
// by ID

G=gap.SmallGroup(480,484);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,142,1356,18822]);
// Polycyclic

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

׿
×
𝔽