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G = D8.D15order 480 = 25·3·5

The non-split extension by D8 of D15 acting via D15/C15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.D15, C60.4D4, C8.5D30, C30.40D8, C40.13D6, C1510SD32, Dic606C2, C24.13D10, C120.10C22, C53(D8.S3), C33(D8.D5), (C3×D8).1D5, (C5×D8).1S3, C153C162C2, (C15×D8).1C2, C2.5(D4⋊D15), C6.18(D4⋊D5), C4.2(C157D4), C10.18(D4⋊S3), C12.18(C5⋊D4), C20.16(C3⋊D4), SmallGroup(480,187)

Series: Derived Chief Lower central Upper central

C1C120 — D8.D15
C1C5C15C30C60C120Dic60 — D8.D15
C15C30C60C120 — D8.D15
C1C2C4C8D8

Generators and relations for D8.D15
 G = < a,b,c,d | a8=b2=c15=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

8C2
4C22
60C4
8C6
8C10
2D4
30Q8
4C2×C6
20Dic3
4C2×C10
12Dic5
8C30
15Q16
15C16
2C3×D4
10Dic6
2C5×D4
6Dic10
4Dic15
4C2×C30
15SD32
5C3⋊C16
5Dic12
3Dic20
3C52C16
2D4×C15
2Dic30
5D8.S3
3D8.D5

Smallest permutation representation of D8.D15
On 240 points
Generators in S240
(1 103 35 84 17 118 50 73)(2 104 36 85 18 119 51 74)(3 105 37 86 19 120 52 75)(4 91 38 87 20 106 53 61)(5 92 39 88 21 107 54 62)(6 93 40 89 22 108 55 63)(7 94 41 90 23 109 56 64)(8 95 42 76 24 110 57 65)(9 96 43 77 25 111 58 66)(10 97 44 78 26 112 59 67)(11 98 45 79 27 113 60 68)(12 99 31 80 28 114 46 69)(13 100 32 81 29 115 47 70)(14 101 33 82 30 116 48 71)(15 102 34 83 16 117 49 72)(121 190 174 234 143 199 157 220)(122 191 175 235 144 200 158 221)(123 192 176 236 145 201 159 222)(124 193 177 237 146 202 160 223)(125 194 178 238 147 203 161 224)(126 195 179 239 148 204 162 225)(127 181 180 240 149 205 163 211)(128 182 166 226 150 206 164 212)(129 183 167 227 136 207 165 213)(130 184 168 228 137 208 151 214)(131 185 169 229 138 209 152 215)(132 186 170 230 139 210 153 216)(133 187 171 231 140 196 154 217)(134 188 172 232 141 197 155 218)(135 189 173 233 142 198 156 219)
(1 73)(2 74)(3 75)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(121 157)(122 158)(123 159)(124 160)(125 161)(126 162)(127 163)(128 164)(129 165)(130 151)(131 152)(132 153)(133 154)(134 155)(135 156)(136 167)(137 168)(138 169)(139 170)(140 171)(141 172)(142 173)(143 174)(144 175)(145 176)(146 177)(147 178)(148 179)(149 180)(150 166)(181 205)(182 206)(183 207)(184 208)(185 209)(186 210)(187 196)(188 197)(189 198)(190 199)(191 200)(192 201)(193 202)(194 203)(195 204)
(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)
(1 150 17 128)(2 149 18 127)(3 148 19 126)(4 147 20 125)(5 146 21 124)(6 145 22 123)(7 144 23 122)(8 143 24 121)(9 142 25 135)(10 141 26 134)(11 140 27 133)(12 139 28 132)(13 138 29 131)(14 137 30 130)(15 136 16 129)(31 170 46 153)(32 169 47 152)(33 168 48 151)(34 167 49 165)(35 166 50 164)(36 180 51 163)(37 179 52 162)(38 178 53 161)(39 177 54 160)(40 176 55 159)(41 175 56 158)(42 174 57 157)(43 173 58 156)(44 172 59 155)(45 171 60 154)(61 203 87 194)(62 202 88 193)(63 201 89 192)(64 200 90 191)(65 199 76 190)(66 198 77 189)(67 197 78 188)(68 196 79 187)(69 210 80 186)(70 209 81 185)(71 208 82 184)(72 207 83 183)(73 206 84 182)(74 205 85 181)(75 204 86 195)(91 238 106 224)(92 237 107 223)(93 236 108 222)(94 235 109 221)(95 234 110 220)(96 233 111 219)(97 232 112 218)(98 231 113 217)(99 230 114 216)(100 229 115 215)(101 228 116 214)(102 227 117 213)(103 226 118 212)(104 240 119 211)(105 239 120 225)

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

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

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

60 conjugacy classes

class 1 2A2B 3 4A4B5A5B6A6B6C8A8B10A10B10C10D10E10F 12 15A15B15C15D16A16B16C16D20A20B24A24B30A30B30C30D30E···30L40A40B40C40D60A60B60C60D120A···120H
order1223445566688101010101010121515151516161616202024243030303030···304040404060606060120···120
size1182212022288222288884222230303030444422228···8444444444···4

60 irreducible representations

dim1111222222222222444444
type++++++++++++++--+-
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4D15SD32C5⋊D4D30C157D4D4⋊S3D4⋊D5D8.S3D8.D5D4⋊D15D8.D15
kernelD8.D15C153C16Dic60C15×D8C5×D8C60C3×D8C40C30C24C20D8C15C12C8C4C10C6C5C3C2C1
# reps1111112122244448122448

Matrix representation of D8.D15 in GL4(𝔽241) generated by

240000
024000
000168
0020822
,
240000
0100
000168
00330
,
100000
09400
0010
0001
,
09400
100000
006292
00118179
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,0,208,0,0,168,22],[240,0,0,0,0,1,0,0,0,0,0,33,0,0,168,0],[100,0,0,0,0,94,0,0,0,0,1,0,0,0,0,1],[0,100,0,0,94,0,0,0,0,0,62,118,0,0,92,179] >;

D8.D15 in GAP, Magma, Sage, TeX

D_8.D_{15}
% in TeX

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

G:=SmallGroup(480,187);
// by ID

G=gap.SmallGroup(480,187);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,254,135,142,675,346,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^15=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D8.D15 in TeX

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