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## G = Dic15⋊19D4order 480 = 25·3·5

### 1st semidirect product of Dic15 and D4 acting through Inn(Dic15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic15⋊19D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C2×C15⋊7D4 — Dic15⋊19D4
 Lower central C15 — C30 — Dic15⋊19D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic1519D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1028 in 188 conjugacy classes, 65 normal (47 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], Dic3 [×5], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×5], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×S3 [×2], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×C6, D15 [×2], C30 [×3], C30 [×2], C4×D4, C4×D5 [×2], C2×Dic5 [×5], C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, Dic15 [×4], Dic15, C60 [×2], D30 [×2], D30 [×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, Dic34D4, C4×D15 [×2], C2×Dic15 [×3], C2×Dic15 [×2], C157D4 [×4], C2×C60 [×2], C22×D15, C22×C30, Dic54D4, C4×Dic15, C30.4Q8, D303C4, C15×C22⋊C4, C2×C4×D15, C22×Dic15, C2×C157D4, Dic1519D4
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, D15, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4, D42S3, D30 [×3], C2×C4×D5, D4×D5, D42D5, Dic34D4, C4×D15 [×2], C22×D15, Dic54D4, C2×C4×D15, D4×D15, D42D15, Dic1519D4

Smallest permutation representation of Dic1519D4
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)
(1 142 16 127)(2 141 17 126)(3 140 18 125)(4 139 19 124)(5 138 20 123)(6 137 21 122)(7 136 22 121)(8 135 23 150)(9 134 24 149)(10 133 25 148)(11 132 26 147)(12 131 27 146)(13 130 28 145)(14 129 29 144)(15 128 30 143)(31 240 46 225)(32 239 47 224)(33 238 48 223)(34 237 49 222)(35 236 50 221)(36 235 51 220)(37 234 52 219)(38 233 53 218)(39 232 54 217)(40 231 55 216)(41 230 56 215)(42 229 57 214)(43 228 58 213)(44 227 59 212)(45 226 60 211)(61 190 76 205)(62 189 77 204)(63 188 78 203)(64 187 79 202)(65 186 80 201)(66 185 81 200)(67 184 82 199)(68 183 83 198)(69 182 84 197)(70 181 85 196)(71 210 86 195)(72 209 87 194)(73 208 88 193)(74 207 89 192)(75 206 90 191)(91 156 106 171)(92 155 107 170)(93 154 108 169)(94 153 109 168)(95 152 110 167)(96 151 111 166)(97 180 112 165)(98 179 113 164)(99 178 114 163)(100 177 115 162)(101 176 116 161)(102 175 117 160)(103 174 118 159)(104 173 119 158)(105 172 120 157)
(1 189 112 218)(2 188 113 217)(3 187 114 216)(4 186 115 215)(5 185 116 214)(6 184 117 213)(7 183 118 212)(8 182 119 211)(9 181 120 240)(10 210 91 239)(11 209 92 238)(12 208 93 237)(13 207 94 236)(14 206 95 235)(15 205 96 234)(16 204 97 233)(17 203 98 232)(18 202 99 231)(19 201 100 230)(20 200 101 229)(21 199 102 228)(22 198 103 227)(23 197 104 226)(24 196 105 225)(25 195 106 224)(26 194 107 223)(27 193 108 222)(28 192 109 221)(29 191 110 220)(30 190 111 219)(31 149 70 172)(32 148 71 171)(33 147 72 170)(34 146 73 169)(35 145 74 168)(36 144 75 167)(37 143 76 166)(38 142 77 165)(39 141 78 164)(40 140 79 163)(41 139 80 162)(42 138 81 161)(43 137 82 160)(44 136 83 159)(45 135 84 158)(46 134 85 157)(47 133 86 156)(48 132 87 155)(49 131 88 154)(50 130 89 153)(51 129 90 152)(52 128 61 151)(53 127 62 180)(54 126 63 179)(55 125 64 178)(56 124 65 177)(57 123 66 176)(58 122 67 175)(59 121 68 174)(60 150 69 173)
(1 218)(2 219)(3 220)(4 221)(5 222)(6 223)(7 224)(8 225)(9 226)(10 227)(11 228)(12 229)(13 230)(14 231)(15 232)(16 233)(17 234)(18 235)(19 236)(20 237)(21 238)(22 239)(23 240)(24 211)(25 212)(26 213)(27 214)(28 215)(29 216)(30 217)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 179)(62 180)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 161)(74 162)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)(89 177)(90 178)(91 198)(92 199)(93 200)(94 201)(95 202)(96 203)(97 204)(98 205)(99 206)(100 207)(101 208)(102 209)(103 210)(104 181)(105 182)(106 183)(107 184)(108 185)(109 186)(110 187)(111 188)(112 189)(113 190)(114 191)(115 192)(116 193)(117 194)(118 195)(119 196)(120 197)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 30 30 2 2 2 2 2 15 15 15 15 30 30 30 30 2 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 4 4 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 C4○D4 D10 D10 C4×S3 D15 C4×D5 D30 D30 C4×D15 S3×D4 D4⋊2S3 D4×D5 D4⋊2D5 D4×D15 D4⋊2D15 kernel Dic15⋊19D4 C4×Dic15 C30.4Q8 D30⋊3C4 C15×C22⋊C4 C2×C4×D15 C22×Dic15 C2×C15⋊7D4 C15⋊7D4 C5×C22⋊C4 Dic15 C3×C22⋊C4 C2×C20 C22×C10 C30 C2×C12 C22×C6 C2×C10 C22⋊C4 C2×C6 C2×C4 C23 C22 C10 C10 C6 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 2 1 2 4 2 4 4 8 8 4 16 1 1 2 2 4 4

Matrix representation of Dic1519D4 in GL6(𝔽61)

 17 60 0 0 0 0 45 1 0 0 0 0 0 0 33 24 0 0 0 0 37 14 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 57 46 0 0 0 0 54 4 0 0 0 0 0 0 60 0 0 0 0 0 44 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 17 18 0 0 0 0 45 44 0 0 0 0 0 0 60 0 0 0 0 0 44 1 0 0 0 0 0 0 60 12 0 0 0 0 10 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 12 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [17,45,0,0,0,0,60,1,0,0,0,0,0,0,33,37,0,0,0,0,24,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,54,0,0,0,0,46,4,0,0,0,0,0,0,60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[17,45,0,0,0,0,18,44,0,0,0,0,0,0,60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,10,0,0,0,0,12,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,12,1] >;

Dic1519D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_{19}D_4
% in TeX

G:=Group("Dic15:19D4");
// GroupNames label

G:=SmallGroup(480,846);
// by ID

G=gap.SmallGroup(480,846);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,219,58,2693,18822]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽