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## G = Dic15⋊13D4order 480 = 25·3·5

### 3rd semidirect product of Dic15 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic15⋊13D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — Dic15⋊13D4
 Lower central C15 — C30 — Dic15⋊13D4
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic1513D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=a11, dad=a19, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1036 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 [×4], C10 [×3], Dic3 [×2], Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×3], C20 [×4], D10 [×2], D10 [×6], C2×C10, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C2×C12, C22×S3, C22×C6, C3×D5 [×2], D15 [×2], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5 [×2], C6×D5 [×2], D30 [×2], D30 [×2], C2×C30, C4×Dic5, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, Dic34D4, D5×Dic3 [×2], C3⋊D20 [×4], C6×Dic5, C10×Dic3 [×2], C4×D15 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D208C4, Dic3×Dic5, D304C4, C3×D10⋊C4, C5×Dic3⋊C4, C2×D5×Dic3, C2×C3⋊D20, C2×C4×D15, Dic1513D4
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, Q82D5, Dic34D4, C2×S3×D5, D208C4, D20⋊S3, C4×S3×D5, D10⋊D6, Dic1513D4

Smallest permutation representation of Dic1513D4
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)
(1 177 16 162)(2 176 17 161)(3 175 18 160)(4 174 19 159)(5 173 20 158)(6 172 21 157)(7 171 22 156)(8 170 23 155)(9 169 24 154)(10 168 25 153)(11 167 26 152)(12 166 27 151)(13 165 28 180)(14 164 29 179)(15 163 30 178)(31 129 46 144)(32 128 47 143)(33 127 48 142)(34 126 49 141)(35 125 50 140)(36 124 51 139)(37 123 52 138)(38 122 53 137)(39 121 54 136)(40 150 55 135)(41 149 56 134)(42 148 57 133)(43 147 58 132)(44 146 59 131)(45 145 60 130)(61 194 76 209)(62 193 77 208)(63 192 78 207)(64 191 79 206)(65 190 80 205)(66 189 81 204)(67 188 82 203)(68 187 83 202)(69 186 84 201)(70 185 85 200)(71 184 86 199)(72 183 87 198)(73 182 88 197)(74 181 89 196)(75 210 90 195)(91 240 106 225)(92 239 107 224)(93 238 108 223)(94 237 109 222)(95 236 110 221)(96 235 111 220)(97 234 112 219)(98 233 113 218)(99 232 114 217)(100 231 115 216)(101 230 116 215)(102 229 117 214)(103 228 118 213)(104 227 119 212)(105 226 120 211)
(1 218 45 202)(2 229 46 183)(3 240 47 194)(4 221 48 205)(5 232 49 186)(6 213 50 197)(7 224 51 208)(8 235 52 189)(9 216 53 200)(10 227 54 181)(11 238 55 192)(12 219 56 203)(13 230 57 184)(14 211 58 195)(15 222 59 206)(16 233 60 187)(17 214 31 198)(18 225 32 209)(19 236 33 190)(20 217 34 201)(21 228 35 182)(22 239 36 193)(23 220 37 204)(24 231 38 185)(25 212 39 196)(26 223 40 207)(27 234 41 188)(28 215 42 199)(29 226 43 210)(30 237 44 191)(61 160 91 128)(62 171 92 139)(63 152 93 150)(64 163 94 131)(65 174 95 142)(66 155 96 123)(67 166 97 134)(68 177 98 145)(69 158 99 126)(70 169 100 137)(71 180 101 148)(72 161 102 129)(73 172 103 140)(74 153 104 121)(75 164 105 132)(76 175 106 143)(77 156 107 124)(78 167 108 135)(79 178 109 146)(80 159 110 127)(81 170 111 138)(82 151 112 149)(83 162 113 130)(84 173 114 141)(85 154 115 122)(86 165 116 133)(87 176 117 144)(88 157 118 125)(89 168 119 136)(90 179 120 147)
(1 45)(2 34)(3 53)(4 42)(5 31)(6 50)(7 39)(8 58)(9 47)(10 36)(11 55)(12 44)(13 33)(14 52)(15 41)(16 60)(17 49)(18 38)(19 57)(20 46)(21 35)(22 54)(23 43)(24 32)(25 51)(26 40)(27 59)(28 48)(29 37)(30 56)(61 85)(62 74)(64 82)(65 71)(66 90)(67 79)(69 87)(70 76)(72 84)(75 81)(77 89)(80 86)(91 115)(92 104)(94 112)(95 101)(96 120)(97 109)(99 117)(100 106)(102 114)(105 111)(107 119)(110 116)(121 171)(122 160)(123 179)(124 168)(125 157)(126 176)(127 165)(128 154)(129 173)(130 162)(131 151)(132 170)(133 159)(134 178)(135 167)(136 156)(137 175)(138 164)(139 153)(140 172)(141 161)(142 180)(143 169)(144 158)(145 177)(146 166)(147 155)(148 174)(149 163)(150 152)(181 193)(183 201)(184 190)(185 209)(186 198)(188 206)(189 195)(191 203)(194 200)(196 208)(199 205)(204 210)(211 235)(212 224)(214 232)(215 221)(216 240)(217 229)(219 237)(220 226)(222 234)(225 231)(227 239)(230 236)

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

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

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

66 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 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H 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 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 30 30 2 2 2 6 6 6 6 10 10 15 15 15 15 2 2 2 2 2 20 20 2 ··· 2 4 4 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 D6 C4○D4 D10 D10 C4×S3 C4×D5 S3×D4 D4⋊2S3 S3×D5 D4×D5 Q8⋊2D5 C2×S3×D5 D20⋊S3 C4×S3×D5 D10⋊D6 kernel Dic15⋊13D4 Dic3×Dic5 D30⋊4C4 C3×D10⋊C4 C5×Dic3⋊C4 C2×D5×Dic3 C2×C3⋊D20 C2×C4×D15 C3⋊D20 D10⋊C4 Dic15 Dic3⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 D10 Dic3 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 1 1 2 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of Dic1513D4 in GL6(𝔽61)

 2 15 0 0 0 0 12 60 0 0 0 0 0 0 43 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 21 0 0 0 0 37 25 0 0 0 0 0 0 18 43 0 0 0 0 1 43 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 31 13 0 0 0 0 20 30 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 6 0 0 0 0 20 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 43 18 0 0 0 0 60 18 0 0 0 0 0 0 60 0 0 0 0 0 20 1

G:=sub<GL(6,GF(61))| [2,12,0,0,0,0,15,60,0,0,0,0,0,0,43,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,37,0,0,0,0,21,25,0,0,0,0,0,0,18,1,0,0,0,0,43,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[31,20,0,0,0,0,13,30,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,20,0,0,0,0,6,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,60,0,0,0,0,18,18,0,0,0,0,0,0,60,20,0,0,0,0,0,1] >;

Dic1513D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_{13}D_4
% in TeX

G:=Group("Dic15:13D4");
// GroupNames label

G:=SmallGroup(480,472);
// by ID

G=gap.SmallGroup(480,472);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,100,1356,18822]);
// 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=a^11,d*a*d=a^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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