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G = D83D15order 480 = 25·3·5

The semidirect product of D8 and D15 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D83D15, C8.8D30, D4.1D30, C40.22D6, Dic607C2, D30.17D4, C24.22D10, C60.66C23, C120.13C22, Dic15.49D4, Dic30.22C22, (C5×D8)⋊3S3, (C3×D8)⋊3D5, (C8×D15)⋊2C2, (C15×D8)⋊3C2, C1528(C4○D8), C54(D83S3), C34(D83D5), (C5×D4).13D6, C2.17(D4×D15), C6.110(D4×D5), D42D159C2, D4.D1510C2, (C3×D4).13D10, C10.112(S3×D4), C30.317(C2×D4), C4.3(C22×D15), C20.104(C22×S3), C153C8.31C22, (C4×D15).43C22, (D4×C15).20C22, C12.104(C22×D5), SmallGroup(480,877)

Series: Derived Chief Lower central Upper central

C1C60 — D83D15
C1C5C15C30C60C4×D15D42D15 — D83D15
C15C30C60 — D83D15
C1C2C4D8

Generators and relations for D83D15
 G = < a,b,c,d | a8=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 724 in 124 conjugacy classes, 41 normal (27 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3, C6, C6 [×2], C8, C8, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], D5, C10, C10 [×2], Dic3 [×3], C12, D6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×3], C20, D10, C2×C10 [×2], C3⋊C8, C24, Dic6 [×2], C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4 [×2], D15, C30, C30 [×2], C4○D8, C52C8, C40, Dic10 [×2], C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4 [×2], S3×C8, Dic12, D4.S3 [×2], C3×D8, D42S3 [×2], Dic15, Dic15 [×2], C60, D30, C2×C30 [×2], C8×D5, Dic20, D4.D5 [×2], C5×D8, D42D5 [×2], D83S3, C153C8, C120, Dic30 [×2], C4×D15, C2×Dic15 [×2], C157D4 [×2], D4×C15 [×2], D83D5, C8×D15, Dic60, D4.D15 [×2], C15×D8, D42D15 [×2], D83D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D15, C4○D8, C22×D5, S3×D4, D30 [×3], D4×D5, D83S3, C22×D15, D83D5, D4×D15, D83D15

Smallest permutation representation of D83D15
On 240 points
Generators in S240
(1 223 43 208 17 234 54 185)(2 224 44 209 18 235 55 186)(3 225 45 210 19 236 56 187)(4 211 31 196 20 237 57 188)(5 212 32 197 21 238 58 189)(6 213 33 198 22 239 59 190)(7 214 34 199 23 240 60 191)(8 215 35 200 24 226 46 192)(9 216 36 201 25 227 47 193)(10 217 37 202 26 228 48 194)(11 218 38 203 27 229 49 195)(12 219 39 204 28 230 50 181)(13 220 40 205 29 231 51 182)(14 221 41 206 30 232 52 183)(15 222 42 207 16 233 53 184)(61 169 106 140 87 154 98 128)(62 170 107 141 88 155 99 129)(63 171 108 142 89 156 100 130)(64 172 109 143 90 157 101 131)(65 173 110 144 76 158 102 132)(66 174 111 145 77 159 103 133)(67 175 112 146 78 160 104 134)(68 176 113 147 79 161 105 135)(69 177 114 148 80 162 91 121)(70 178 115 149 81 163 92 122)(71 179 116 150 82 164 93 123)(72 180 117 136 83 165 94 124)(73 166 118 137 84 151 95 125)(74 167 119 138 85 152 96 126)(75 168 120 139 86 153 97 127)
(1 125)(2 126)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 134)(11 135)(12 121)(13 122)(14 123)(15 124)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 154)(32 155)(33 156)(34 157)(35 158)(36 159)(37 160)(38 161)(39 162)(40 163)(41 164)(42 165)(43 151)(44 152)(45 153)(46 173)(47 174)(48 175)(49 176)(50 177)(51 178)(52 179)(53 180)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 188)(62 189)(63 190)(64 191)(65 192)(66 193)(67 194)(68 195)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 200)(77 201)(78 202)(79 203)(80 204)(81 205)(82 206)(83 207)(84 208)(85 209)(86 210)(87 196)(88 197)(89 198)(90 199)(91 219)(92 220)(93 221)(94 222)(95 223)(96 224)(97 225)(98 211)(99 212)(100 213)(101 214)(102 215)(103 216)(104 217)(105 218)(106 237)(107 238)(108 239)(109 240)(110 226)(111 227)(112 228)(113 229)(114 230)(115 231)(116 232)(117 233)(118 234)(119 235)(120 236)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 106)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(121 140)(122 139)(123 138)(124 137)(125 136)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(151 180)(152 179)(153 178)(154 177)(155 176)(156 175)(157 174)(158 173)(159 172)(160 171)(161 170)(162 169)(163 168)(164 167)(165 166)(181 188)(182 187)(183 186)(184 185)(189 195)(190 194)(191 193)(196 204)(197 203)(198 202)(199 201)(205 210)(206 209)(207 208)(211 219)(212 218)(213 217)(214 216)(220 225)(221 224)(222 223)(227 240)(228 239)(229 238)(230 237)(231 236)(232 235)(233 234)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C8A8B8C8D10A10B10C10D10E10F 12 15A15B15C15D20A20B24A24B30A30B30C30D30E···30L40A40B40C40D60A60B60C60D120A···120H
order122223444445566688881010101010101215151515202024243030303030···304040404060606060120···120
size11443022151560602228822303022888842222444422228···8444444444···4

63 irreducible representations

dim111111222222222222444444
type+++++++++++++++++++--+-
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D15C4○D8D30D30S3×D4D4×D5D83S3D83D5D4×D15D83D15
kernelD83D15C8×D15Dic60D4.D15C15×D8D42D15C5×D8Dic15D30C3×D8C40C5×D4C24C3×D4D8C15C8D4C10C6C5C3C2C1
# reps111212111212244448122448

Matrix representation of D83D15 in GL4(𝔽241) generated by

240000
024000
002110
0008
,
240000
024000
0008
002110
,
9411000
13116100
0010
0001
,
9411000
8414700
0010
000240
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,211,0,0,0,0,8],[240,0,0,0,0,240,0,0,0,0,0,211,0,0,8,0],[94,131,0,0,110,161,0,0,0,0,1,0,0,0,0,1],[94,84,0,0,110,147,0,0,0,0,1,0,0,0,0,240] >;

D83D15 in GAP, Magma, Sage, TeX

D_8\rtimes_3D_{15}
% in TeX

G:=Group("D8:3D15");
// GroupNames label

G:=SmallGroup(480,877);
// by ID

G=gap.SmallGroup(480,877);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,135,346,185,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^15=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽