metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D21⋊C4, D42.C2, C14.3D6, C6.3D14, Dic3⋊2D7, Dic7⋊2S3, C42.3C22, C7⋊1(C4×S3), C3⋊1(C4×D7), C21⋊3(C2×C4), C2.3(S3×D7), (C7×Dic3)⋊2C2, (C3×Dic7)⋊2C2, SmallGroup(168,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — D21⋊C4 |
Generators and relations for D21⋊C4
G = < a,b,c | a21=b2=c4=1, bab=a-1, cac-1=a8, cbc-1=a7b >
Character table of D21⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 7A | 7B | 7C | 12A | 12B | 14A | 14B | 14C | 21A | 21B | 21C | 28A | 28B | 28C | 28D | 28E | 28F | 42A | 42B | 42C | |
| size | 1 | 1 | 21 | 21 | 2 | 3 | 3 | 7 | 7 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -i | i | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | i | i | -i | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | i | i | -i | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -i | i | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 2 | 2 | -1 | 2 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -2 | -2 | -1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ12 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ16 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 2i | -2i | 1 | 2 | 2 | 2 | i | -i | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | complex lifted from C4×S3 |
| ρ18 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -2i | 2i | 1 | 2 | 2 | 2 | -i | i | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | complex lifted from C4×S3 |
| ρ19 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ4ζ74+ζ4ζ73 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | ζ43ζ76+ζ43ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | complex lifted from C4×D7 |
| ρ20 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ43ζ75+ζ43ζ72 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | ζ4ζ74+ζ4ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | complex lifted from C4×D7 |
| ρ21 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 0 | 0 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ43ζ74+ζ43ζ73 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | ζ4ζ76+ζ4ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | complex lifted from C4×D7 |
| ρ22 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ43ζ76+ζ43ζ7 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | ζ4ζ75+ζ4ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | complex lifted from C4×D7 |
| ρ23 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 0 | 0 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ4ζ75+ζ4ζ72 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | ζ43ζ74+ζ43ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | complex lifted from C4×D7 |
| ρ24 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 0 | 0 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ4ζ76+ζ4ζ7 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | ζ43ζ75+ζ43ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | complex lifted from C4×D7 |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | -2ζ75-2ζ72 | -2ζ76-2ζ7 | -2ζ74-2ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | orthogonal faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | -2ζ76-2ζ7 | -2ζ74-2ζ73 | -2ζ75-2ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | orthogonal faithful, Schur index 2 |
| ρ27 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | -2ζ74-2ζ73 | -2ζ75-2ζ72 | -2ζ76-2ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | orthogonal faithful, Schur index 2 |
| ρ29 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
| ρ30 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
►On 84 points
Generators in S84
(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)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 31)(23 30)(24 29)(25 28)(26 27)(32 42)(33 41)(34 40)(35 39)(36 38)(43 45)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(64 74)(65 73)(66 72)(67 71)(68 70)(75 84)(76 83)(77 82)(78 81)(79 80)
(1 80 27 55)(2 67 28 63)(3 75 29 50)(4 83 30 58)(5 70 31 45)(6 78 32 53)(7 65 33 61)(8 73 34 48)(9 81 35 56)(10 68 36 43)(11 76 37 51)(12 84 38 59)(13 71 39 46)(14 79 40 54)(15 66 41 62)(16 74 42 49)(17 82 22 57)(18 69 23 44)(19 77 24 52)(20 64 25 60)(21 72 26 47)
G:=sub<Sym(84)| (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,31)(23,30)(24,29)(25,28)(26,27)(32,42)(33,41)(34,40)(35,39)(36,38)(43,45)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(64,74)(65,73)(66,72)(67,71)(68,70)(75,84)(76,83)(77,82)(78,81)(79,80), (1,80,27,55)(2,67,28,63)(3,75,29,50)(4,83,30,58)(5,70,31,45)(6,78,32,53)(7,65,33,61)(8,73,34,48)(9,81,35,56)(10,68,36,43)(11,76,37,51)(12,84,38,59)(13,71,39,46)(14,79,40,54)(15,66,41,62)(16,74,42,49)(17,82,22,57)(18,69,23,44)(19,77,24,52)(20,64,25,60)(21,72,26,47)>;
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,31)(23,30)(24,29)(25,28)(26,27)(32,42)(33,41)(34,40)(35,39)(36,38)(43,45)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(64,74)(65,73)(66,72)(67,71)(68,70)(75,84)(76,83)(77,82)(78,81)(79,80), (1,80,27,55)(2,67,28,63)(3,75,29,50)(4,83,30,58)(5,70,31,45)(6,78,32,53)(7,65,33,61)(8,73,34,48)(9,81,35,56)(10,68,36,43)(11,76,37,51)(12,84,38,59)(13,71,39,46)(14,79,40,54)(15,66,41,62)(16,74,42,49)(17,82,22,57)(18,69,23,44)(19,77,24,52)(20,64,25,60)(21,72,26,47) );
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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,31),(23,30),(24,29),(25,28),(26,27),(32,42),(33,41),(34,40),(35,39),(36,38),(43,45),(46,63),(47,62),(48,61),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(64,74),(65,73),(66,72),(67,71),(68,70),(75,84),(76,83),(77,82),(78,81),(79,80)], [(1,80,27,55),(2,67,28,63),(3,75,29,50),(4,83,30,58),(5,70,31,45),(6,78,32,53),(7,65,33,61),(8,73,34,48),(9,81,35,56),(10,68,36,43),(11,76,37,51),(12,84,38,59),(13,71,39,46),(14,79,40,54),(15,66,41,62),(16,74,42,49),(17,82,22,57),(18,69,23,44),(19,77,24,52),(20,64,25,60),(21,72,26,47)]])
D21⋊C4 is a maximal subgroup of
D84⋊C2 D21⋊Q8 D14.D6 C4×S3×D7 Dic7.D6 Dic3.D14 D6⋊D14
D21⋊C4 is a maximal quotient of D21⋊C8 D42.C4 Dic3×Dic7 D42⋊C4 Dic21⋊C4
Matrix representation of D21⋊C4 ►in GL4(𝔽337) generated by
| 0 | 1 | 0 | 0 |
| 336 | 336 | 0 | 0 |
| 0 | 0 | 34 | 194 |
| 0 | 0 | 178 | 193 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 228 |
| 0 | 0 | 34 | 0 |
| 336 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 148 | 0 |
| 0 | 0 | 0 | 148 |
G:=sub<GL(4,GF(337))| [0,336,0,0,1,336,0,0,0,0,34,178,0,0,194,193],[0,1,0,0,1,0,0,0,0,0,0,34,0,0,228,0],[336,1,0,0,0,1,0,0,0,0,148,0,0,0,0,148] >;
D21⋊C4 in GAP, Magma, Sage, TeX
D_{21}\rtimes C_4 % in TeX
G:=Group("D21:C4"); // GroupNames label
G:=SmallGroup(168,14);
// by ID
G=gap.SmallGroup(168,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,20,26,168,3604]);
// Polycyclic
G:=Group<a,b,c|a^21=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^8,c*b*c^-1=a^7*b>;
// generators/relations
Export
Subgroup lattice of D21⋊C4 in TeX
Character table of D21⋊C4 in TeX