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G = D21order 42 = 2·3·7

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D21, C7⋊S3, C3⋊D7, C211C2, sometimes denoted D42 or Dih21 or Dih42, SmallGroup(42,5)

Series: Derived Chief Lower central Upper central

C1C21 — D21
C1C7C21 — D21
C21 — D21
C1

Generators and relations for D21
 G = < a,b | a21=b2=1, bab=a-1 >

21C2
7S3
3D7

Character table of D21

 class 1237A7B7C21A21B21C21D21E21F
 size 1212222222222
ρ1111111111111    trivial
ρ21-11111111111    linear of order 2
ρ320-1222-1-1-1-1-1-1    orthogonal lifted from S3
ρ4202ζ7473ζ767ζ7572ζ7473ζ767ζ767ζ7473ζ7572ζ7572    orthogonal lifted from D7
ρ5202ζ7572ζ7473ζ767ζ7572ζ7473ζ7473ζ7572ζ767ζ767    orthogonal lifted from D7
ρ6202ζ767ζ7572ζ7473ζ767ζ7572ζ7572ζ767ζ7473ζ7473    orthogonal lifted from D7
ρ720-1ζ7572ζ7473ζ767ζ3ζ753ζ727232ζ7432ζ7374ζ32ζ7432ζ7373ζ32ζ7532ζ72723ζ763ζ776ζ3ζ763ζ77    orthogonal faithful
ρ820-1ζ767ζ7572ζ74733ζ763ζ776ζ3ζ753ζ7272ζ32ζ7532ζ7272ζ3ζ763ζ77ζ32ζ7432ζ737332ζ7432ζ7374    orthogonal faithful
ρ920-1ζ7473ζ767ζ7572ζ32ζ7432ζ73733ζ763ζ776ζ3ζ763ζ7732ζ7432ζ7374ζ32ζ7532ζ7272ζ3ζ753ζ7272    orthogonal faithful
ρ1020-1ζ767ζ7572ζ7473ζ3ζ763ζ77ζ32ζ7532ζ7272ζ3ζ753ζ72723ζ763ζ77632ζ7432ζ7374ζ32ζ7432ζ7373    orthogonal faithful
ρ1120-1ζ7572ζ7473ζ767ζ32ζ7532ζ7272ζ32ζ7432ζ737332ζ7432ζ7374ζ3ζ753ζ7272ζ3ζ763ζ773ζ763ζ776    orthogonal faithful
ρ1220-1ζ7473ζ767ζ757232ζ7432ζ7374ζ3ζ763ζ773ζ763ζ776ζ32ζ7432ζ7373ζ3ζ753ζ7272ζ32ζ7532ζ7272    orthogonal faithful

Permutation representations of D21
On 21 points - transitive group 21T5
Generators in S21
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)

G:=sub<Sym(21)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12)])

G:=TransitiveGroup(21,5);

Matrix representation of D21 in GL2(𝔽41) generated by

4020
2130
,
3035
2011
G:=sub<GL(2,GF(41))| [40,21,20,30],[30,20,35,11] >;

D21 in GAP, Magma, Sage, TeX

D_{21}
% in TeX

G:=Group("D21");
// GroupNames label

G:=SmallGroup(42,5);
// by ID

G=gap.SmallGroup(42,5);
# by ID

G:=PCGroup([3,-2,-3,-7,25,326]);
// Polycyclic

G:=Group<a,b|a^21=b^2=1,b*a*b=a^-1>;
// generators/relations

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