metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊9Q8, C42.175D10, C10.832+ (1+4), C4⋊Q8⋊13D5, C4.19(Q8×D5), C5⋊8(D4⋊3Q8), C20.56(C2×Q8), C4⋊C4.220D10, (C4×D20).27C2, (C2×Q8).87D10, D10.24(C2×Q8), D10⋊2Q8⋊44C2, D10⋊3Q8⋊37C2, (C4×Dic10)⋊53C2, C4.Dic10⋊44C2, D20⋊8C4.14C2, C20.137(C4○D4), C10.50(C22×Q8), (C4×C20).215C22, (C2×C10).274C24, (C2×C20).107C23, C4.40(Q8⋊2D5), C2.87(D4⋊6D10), (C2×D20).281C22, C4⋊Dic5.253C22, (Q8×C10).141C22, C22.295(C23×D5), (C2×Dic5).145C23, (C4×Dic5).171C22, C10.D4.62C22, (C22×D5).245C23, D10⋊C4.153C22, (C2×Dic10).311C22, (D5×C4⋊C4)⋊45C2, C2.33(C2×Q8×D5), (C5×C4⋊Q8)⋊16C2, C10.122(C2×C4○D4), C2.30(C2×Q8⋊2D5), (C2×C4×D5).156C22, (C5×C4⋊C4).217C22, (C2×C4).220(C22×D5), SmallGroup(320,1402)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 774 in 228 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22, C22 [×8], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×6], C20 [×4], C20 [×5], D10 [×4], D10 [×4], C2×C10, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×8], D20 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], D4⋊3Q8, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×6], C2×D20, Q8×C10 [×2], C4×Dic10, C4×D20, C4.Dic10 [×2], D5×C4⋊C4 [×2], D20⋊8C4 [×2], D10⋊2Q8 [×2], D10⋊3Q8 [×4], C5×C4⋊Q8, D20⋊9Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C22×D5 [×7], D4⋊3Q8, Q8×D5 [×2], Q8⋊2D5 [×2], C23×D5, D4⋊6D10, C2×Q8×D5, C2×Q8⋊2D5, D20⋊9Q8
Generators and relations
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 >
►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 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 147)(142 146)(143 145)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)
(1 100 127 152)(2 91 128 143)(3 82 129 154)(4 93 130 145)(5 84 131 156)(6 95 132 147)(7 86 133 158)(8 97 134 149)(9 88 135 160)(10 99 136 151)(11 90 137 142)(12 81 138 153)(13 92 139 144)(14 83 140 155)(15 94 121 146)(16 85 122 157)(17 96 123 148)(18 87 124 159)(19 98 125 150)(20 89 126 141)(21 57 105 62)(22 48 106 73)(23 59 107 64)(24 50 108 75)(25 41 109 66)(26 52 110 77)(27 43 111 68)(28 54 112 79)(29 45 113 70)(30 56 114 61)(31 47 115 72)(32 58 116 63)(33 49 117 74)(34 60 118 65)(35 51 119 76)(36 42 120 67)(37 53 101 78)(38 44 102 69)(39 55 103 80)(40 46 104 71)
(1 26 127 110)(2 27 128 111)(3 28 129 112)(4 29 130 113)(5 30 131 114)(6 31 132 115)(7 32 133 116)(8 33 134 117)(9 34 135 118)(10 35 136 119)(11 36 137 120)(12 37 138 101)(13 38 139 102)(14 39 140 103)(15 40 121 104)(16 21 122 105)(17 22 123 106)(18 23 124 107)(19 24 125 108)(20 25 126 109)(41 89 66 141)(42 90 67 142)(43 91 68 143)(44 92 69 144)(45 93 70 145)(46 94 71 146)(47 95 72 147)(48 96 73 148)(49 97 74 149)(50 98 75 150)(51 99 76 151)(52 100 77 152)(53 81 78 153)(54 82 79 154)(55 83 80 155)(56 84 61 156)(57 85 62 157)(58 86 63 158)(59 87 64 159)(60 88 65 160)
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)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,147)(142,146)(143,145)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155), (1,100,127,152)(2,91,128,143)(3,82,129,154)(4,93,130,145)(5,84,131,156)(6,95,132,147)(7,86,133,158)(8,97,134,149)(9,88,135,160)(10,99,136,151)(11,90,137,142)(12,81,138,153)(13,92,139,144)(14,83,140,155)(15,94,121,146)(16,85,122,157)(17,96,123,148)(18,87,124,159)(19,98,125,150)(20,89,126,141)(21,57,105,62)(22,48,106,73)(23,59,107,64)(24,50,108,75)(25,41,109,66)(26,52,110,77)(27,43,111,68)(28,54,112,79)(29,45,113,70)(30,56,114,61)(31,47,115,72)(32,58,116,63)(33,49,117,74)(34,60,118,65)(35,51,119,76)(36,42,120,67)(37,53,101,78)(38,44,102,69)(39,55,103,80)(40,46,104,71), (1,26,127,110)(2,27,128,111)(3,28,129,112)(4,29,130,113)(5,30,131,114)(6,31,132,115)(7,32,133,116)(8,33,134,117)(9,34,135,118)(10,35,136,119)(11,36,137,120)(12,37,138,101)(13,38,139,102)(14,39,140,103)(15,40,121,104)(16,21,122,105)(17,22,123,106)(18,23,124,107)(19,24,125,108)(20,25,126,109)(41,89,66,141)(42,90,67,142)(43,91,68,143)(44,92,69,144)(45,93,70,145)(46,94,71,146)(47,95,72,147)(48,96,73,148)(49,97,74,149)(50,98,75,150)(51,99,76,151)(52,100,77,152)(53,81,78,153)(54,82,79,154)(55,83,80,155)(56,84,61,156)(57,85,62,157)(58,86,63,158)(59,87,64,159)(60,88,65,160)>;
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)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,147)(142,146)(143,145)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155), (1,100,127,152)(2,91,128,143)(3,82,129,154)(4,93,130,145)(5,84,131,156)(6,95,132,147)(7,86,133,158)(8,97,134,149)(9,88,135,160)(10,99,136,151)(11,90,137,142)(12,81,138,153)(13,92,139,144)(14,83,140,155)(15,94,121,146)(16,85,122,157)(17,96,123,148)(18,87,124,159)(19,98,125,150)(20,89,126,141)(21,57,105,62)(22,48,106,73)(23,59,107,64)(24,50,108,75)(25,41,109,66)(26,52,110,77)(27,43,111,68)(28,54,112,79)(29,45,113,70)(30,56,114,61)(31,47,115,72)(32,58,116,63)(33,49,117,74)(34,60,118,65)(35,51,119,76)(36,42,120,67)(37,53,101,78)(38,44,102,69)(39,55,103,80)(40,46,104,71), (1,26,127,110)(2,27,128,111)(3,28,129,112)(4,29,130,113)(5,30,131,114)(6,31,132,115)(7,32,133,116)(8,33,134,117)(9,34,135,118)(10,35,136,119)(11,36,137,120)(12,37,138,101)(13,38,139,102)(14,39,140,103)(15,40,121,104)(16,21,122,105)(17,22,123,106)(18,23,124,107)(19,24,125,108)(20,25,126,109)(41,89,66,141)(42,90,67,142)(43,91,68,143)(44,92,69,144)(45,93,70,145)(46,94,71,146)(47,95,72,147)(48,96,73,148)(49,97,74,149)(50,98,75,150)(51,99,76,151)(52,100,77,152)(53,81,78,153)(54,82,79,154)(55,83,80,155)(56,84,61,156)(57,85,62,157)(58,86,63,158)(59,87,64,159)(60,88,65,160) );
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),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,60),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(81,83),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(101,103),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(121,127),(122,126),(123,125),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(141,147),(142,146),(143,145),(148,160),(149,159),(150,158),(151,157),(152,156),(153,155)], [(1,100,127,152),(2,91,128,143),(3,82,129,154),(4,93,130,145),(5,84,131,156),(6,95,132,147),(7,86,133,158),(8,97,134,149),(9,88,135,160),(10,99,136,151),(11,90,137,142),(12,81,138,153),(13,92,139,144),(14,83,140,155),(15,94,121,146),(16,85,122,157),(17,96,123,148),(18,87,124,159),(19,98,125,150),(20,89,126,141),(21,57,105,62),(22,48,106,73),(23,59,107,64),(24,50,108,75),(25,41,109,66),(26,52,110,77),(27,43,111,68),(28,54,112,79),(29,45,113,70),(30,56,114,61),(31,47,115,72),(32,58,116,63),(33,49,117,74),(34,60,118,65),(35,51,119,76),(36,42,120,67),(37,53,101,78),(38,44,102,69),(39,55,103,80),(40,46,104,71)], [(1,26,127,110),(2,27,128,111),(3,28,129,112),(4,29,130,113),(5,30,131,114),(6,31,132,115),(7,32,133,116),(8,33,134,117),(9,34,135,118),(10,35,136,119),(11,36,137,120),(12,37,138,101),(13,38,139,102),(14,39,140,103),(15,40,121,104),(16,21,122,105),(17,22,123,106),(18,23,124,107),(19,24,125,108),(20,25,126,109),(41,89,66,141),(42,90,67,142),(43,91,68,143),(44,92,69,144),(45,93,70,145),(46,94,71,146),(47,95,72,147),(48,96,73,148),(49,97,74,149),(50,98,75,150),(51,99,76,151),(52,100,77,152),(53,81,78,153),(54,82,79,154),(55,83,80,155),(56,84,61,156),(57,85,62,157),(58,86,63,158),(59,87,64,159),(60,88,65,160)])
Matrix representation ►G ⊆ 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] >;
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 | C4○D4 | D10 | D10 | D10 | 2+ (1+4) | Q8×D5 | Q8⋊2D5 | D4⋊6D10 |
| kernel | D20⋊9Q8 | C4×Dic10 | C4×D20 | C4.Dic10 | D5×C4⋊C4 | D20⋊8C4 | D10⋊2Q8 | D10⋊3Q8 | C5×C4⋊Q8 | D20 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 4 | 2 | 4 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
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
×
⋊
ℤ
𝔽
○
≀