metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20:9Q8, C42.175D10, C10.832+ 1+4, C4:Q8:13D5, C4.19(Q8xD5), C5:8(D4:3Q8), C20.56(C2xQ8), C4:C4.220D10, (C4xD20).27C2, D10.24(C2xQ8), (C2xQ8).87D10, D10:2Q8:44C2, D10:3Q8:37C2, (C4xDic10):53C2, C4.Dic10:44C2, D20:8C4.14C2, C20.137(C4oD4), C10.50(C22xQ8), (C2xC10).274C24, (C4xC20).215C22, (C2xC20).107C23, C4.40(Q8:2D5), C2.87(D4:6D10), (C2xD20).281C22, C4:Dic5.253C22, (Q8xC10).141C22, C22.295(C23xD5), (C4xDic5).171C22, (C2xDic5).145C23, C10.D4.62C22, (C22xD5).245C23, D10:C4.153C22, (C2xDic10).311C22, (D5xC4:C4):45C2, C2.33(C2xQ8xD5), (C5xC4:Q8):16C2, C10.122(C2xC4oD4), C2.30(C2xQ8:2D5), (C2xC4xD5).156C22, (C5xC4:C4).217C22, (C2xC4).220(C22xD5), SmallGroup(320,1402)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20:9Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=dbd-1=a10b, dcd-1=c-1 >
Subgroups: 774 in 228 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C42, C42, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C2xQ8, C2xQ8, Dic5, C20, C20, D10, D10, C2xC10, C2xC4:C4, C4xD4, C4xQ8, C22:Q8, C42.C2, C4:Q8, Dic10, C4xD5, D20, C2xDic5, C2xC20, C2xC20, C5xQ8, C22xD5, D4:3Q8, C4xDic5, C10.D4, C4:Dic5, C4:Dic5, D10:C4, C4xC20, C5xC4:C4, C2xDic10, C2xC4xD5, C2xD20, Q8xC10, C4xDic10, C4xD20, C4.Dic10, D5xC4:C4, D20:8C4, D10:2Q8, D10:3Q8, C5xC4:Q8, D20:9Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2xQ8, C4oD4, C24, D10, C22xQ8, C2xC4oD4, 2+ 1+4, C22xD5, D4:3Q8, Q8xD5, Q8:2D5, C23xD5, D4:6D10, C2xQ8xD5, C2xQ8:2D5, D20:9Q8
►On 160 points
Generators in S160
(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 85)(82 84)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(114 120)(115 119)(116 118)(121 131)(122 130)(123 129)(124 128)(125 127)(132 140)(133 139)(134 138)(135 137)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)
(1 134 120 155)(2 125 101 146)(3 136 102 157)(4 127 103 148)(5 138 104 159)(6 129 105 150)(7 140 106 141)(8 131 107 152)(9 122 108 143)(10 133 109 154)(11 124 110 145)(12 135 111 156)(13 126 112 147)(14 137 113 158)(15 128 114 149)(16 139 115 160)(17 130 116 151)(18 121 117 142)(19 132 118 153)(20 123 119 144)(21 91 65 56)(22 82 66 47)(23 93 67 58)(24 84 68 49)(25 95 69 60)(26 86 70 51)(27 97 71 42)(28 88 72 53)(29 99 73 44)(30 90 74 55)(31 81 75 46)(32 92 76 57)(33 83 77 48)(34 94 78 59)(35 85 79 50)(36 96 80 41)(37 87 61 52)(38 98 62 43)(39 89 63 54)(40 100 64 45)
(1 80 120 36)(2 61 101 37)(3 62 102 38)(4 63 103 39)(5 64 104 40)(6 65 105 21)(7 66 106 22)(8 67 107 23)(9 68 108 24)(10 69 109 25)(11 70 110 26)(12 71 111 27)(13 72 112 28)(14 73 113 29)(15 74 114 30)(16 75 115 31)(17 76 116 32)(18 77 117 33)(19 78 118 34)(20 79 119 35)(41 134 96 155)(42 135 97 156)(43 136 98 157)(44 137 99 158)(45 138 100 159)(46 139 81 160)(47 140 82 141)(48 121 83 142)(49 122 84 143)(50 123 85 144)(51 124 86 145)(52 125 87 146)(53 126 88 147)(54 127 89 148)(55 128 90 149)(56 129 91 150)(57 130 92 151)(58 131 93 152)(59 132 94 153)(60 133 95 154)
G:=sub<Sym(160)| (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118)(121,131)(122,130)(123,129)(124,128)(125,127)(132,140)(133,139)(134,138)(135,137)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,160)(155,159)(156,158), (1,134,120,155)(2,125,101,146)(3,136,102,157)(4,127,103,148)(5,138,104,159)(6,129,105,150)(7,140,106,141)(8,131,107,152)(9,122,108,143)(10,133,109,154)(11,124,110,145)(12,135,111,156)(13,126,112,147)(14,137,113,158)(15,128,114,149)(16,139,115,160)(17,130,116,151)(18,121,117,142)(19,132,118,153)(20,123,119,144)(21,91,65,56)(22,82,66,47)(23,93,67,58)(24,84,68,49)(25,95,69,60)(26,86,70,51)(27,97,71,42)(28,88,72,53)(29,99,73,44)(30,90,74,55)(31,81,75,46)(32,92,76,57)(33,83,77,48)(34,94,78,59)(35,85,79,50)(36,96,80,41)(37,87,61,52)(38,98,62,43)(39,89,63,54)(40,100,64,45), (1,80,120,36)(2,61,101,37)(3,62,102,38)(4,63,103,39)(5,64,104,40)(6,65,105,21)(7,66,106,22)(8,67,107,23)(9,68,108,24)(10,69,109,25)(11,70,110,26)(12,71,111,27)(13,72,112,28)(14,73,113,29)(15,74,114,30)(16,75,115,31)(17,76,116,32)(18,77,117,33)(19,78,118,34)(20,79,119,35)(41,134,96,155)(42,135,97,156)(43,136,98,157)(44,137,99,158)(45,138,100,159)(46,139,81,160)(47,140,82,141)(48,121,83,142)(49,122,84,143)(50,123,85,144)(51,124,86,145)(52,125,87,146)(53,126,88,147)(54,127,89,148)(55,128,90,149)(56,129,91,150)(57,130,92,151)(58,131,93,152)(59,132,94,153)(60,133,95,154)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118)(121,131)(122,130)(123,129)(124,128)(125,127)(132,140)(133,139)(134,138)(135,137)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,160)(155,159)(156,158), (1,134,120,155)(2,125,101,146)(3,136,102,157)(4,127,103,148)(5,138,104,159)(6,129,105,150)(7,140,106,141)(8,131,107,152)(9,122,108,143)(10,133,109,154)(11,124,110,145)(12,135,111,156)(13,126,112,147)(14,137,113,158)(15,128,114,149)(16,139,115,160)(17,130,116,151)(18,121,117,142)(19,132,118,153)(20,123,119,144)(21,91,65,56)(22,82,66,47)(23,93,67,58)(24,84,68,49)(25,95,69,60)(26,86,70,51)(27,97,71,42)(28,88,72,53)(29,99,73,44)(30,90,74,55)(31,81,75,46)(32,92,76,57)(33,83,77,48)(34,94,78,59)(35,85,79,50)(36,96,80,41)(37,87,61,52)(38,98,62,43)(39,89,63,54)(40,100,64,45), (1,80,120,36)(2,61,101,37)(3,62,102,38)(4,63,103,39)(5,64,104,40)(6,65,105,21)(7,66,106,22)(8,67,107,23)(9,68,108,24)(10,69,109,25)(11,70,110,26)(12,71,111,27)(13,72,112,28)(14,73,113,29)(15,74,114,30)(16,75,115,31)(17,76,116,32)(18,77,117,33)(19,78,118,34)(20,79,119,35)(41,134,96,155)(42,135,97,156)(43,136,98,157)(44,137,99,158)(45,138,100,159)(46,139,81,160)(47,140,82,141)(48,121,83,142)(49,122,84,143)(50,123,85,144)(51,124,86,145)(52,125,87,146)(53,126,88,147)(54,127,89,148)(55,128,90,149)(56,129,91,150)(57,130,92,151)(58,131,93,152)(59,132,94,153)(60,133,95,154) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(36,40),(37,39),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,60),(57,59),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,85),(82,84),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108),(114,120),(115,119),(116,118),(121,131),(122,130),(123,129),(124,128),(125,127),(132,140),(133,139),(134,138),(135,137),(141,153),(142,152),(143,151),(144,150),(145,149),(146,148),(154,160),(155,159),(156,158)], [(1,134,120,155),(2,125,101,146),(3,136,102,157),(4,127,103,148),(5,138,104,159),(6,129,105,150),(7,140,106,141),(8,131,107,152),(9,122,108,143),(10,133,109,154),(11,124,110,145),(12,135,111,156),(13,126,112,147),(14,137,113,158),(15,128,114,149),(16,139,115,160),(17,130,116,151),(18,121,117,142),(19,132,118,153),(20,123,119,144),(21,91,65,56),(22,82,66,47),(23,93,67,58),(24,84,68,49),(25,95,69,60),(26,86,70,51),(27,97,71,42),(28,88,72,53),(29,99,73,44),(30,90,74,55),(31,81,75,46),(32,92,76,57),(33,83,77,48),(34,94,78,59),(35,85,79,50),(36,96,80,41),(37,87,61,52),(38,98,62,43),(39,89,63,54),(40,100,64,45)], [(1,80,120,36),(2,61,101,37),(3,62,102,38),(4,63,103,39),(5,64,104,40),(6,65,105,21),(7,66,106,22),(8,67,107,23),(9,68,108,24),(10,69,109,25),(11,70,110,26),(12,71,111,27),(13,72,112,28),(14,73,113,29),(15,74,114,30),(16,75,115,31),(17,76,116,32),(18,77,117,33),(19,78,118,34),(20,79,119,35),(41,134,96,155),(42,135,97,156),(43,136,98,157),(44,137,99,158),(45,138,100,159),(46,139,81,160),(47,140,82,141),(48,121,83,142),(49,122,84,143),(50,123,85,144),(51,124,86,145),(52,125,87,146),(53,126,88,147),(54,127,89,148),(55,128,90,149),(56,129,91,150),(57,130,92,151),(58,131,93,152),(59,132,94,153),(60,133,95,154)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4oD4 | D10 | D10 | D10 | 2+ 1+4 | Q8xD5 | Q8:2D5 | D4:6D10 |
| kernel | D20:9Q8 | C4xDic10 | C4xD20 | C4.Dic10 | D5xC4:C4 | D20:8C4 | D10:2Q8 | D10:3Q8 | C5xC4:Q8 | D20 | C4:Q8 | C20 | C42 | C4:C4 | C2xQ8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 4 | 2 | 4 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of D20:9Q8 ►in GL6(F41)
| 9 | 23 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 32 | 18 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 7 |
| 0 | 0 | 0 | 0 | 7 | 27 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,23,32,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[32,0,0,0,0,0,18,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,14,7,0,0,0,0,7,27] >;
D20:9Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_9Q_8 % in TeX
G:=Group("D20:9Q8"); // GroupNames label
G:=SmallGroup(320,1402);
// by ID
G=gap.SmallGroup(320,1402);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,570,185,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations