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G = D42order 84 = 22·3·7

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D42, C2×D21, C6⋊D7, C14⋊S3, C72D6, C32D14, C421C2, C212C22, sometimes denoted D84 or Dih42 or Dih84, SmallGroup(84,14)

Series: Derived Chief Lower central Upper central

C1C21 — D42
C1C7C21D21 — D42
C21 — D42
C1C2

Generators and relations for D42
 G = < a,b | a42=b2=1, bab=a-1 >

21C2
21C2
21C22
7S3
7S3
3D7
3D7
7D6
3D14

Character table of D42

 class 12A2B2C367A7B7C14A14B14C21A21B21C21D21E21F42A42B42C42D42E42F
 size 11212122222222222222222222
ρ1111111111111111111111111    trivial
ρ21-11-11-1111-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ31-1-111-1111-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ411-1-111111111111111111111    linear of order 2
ρ52-200-11222-2-2-2-1-1-1-1-1-1111111    orthogonal lifted from D6
ρ62200-1-1222222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ7220022ζ7473ζ7572ζ767ζ7473ζ767ζ7572ζ7572ζ767ζ767ζ7572ζ7473ζ7473ζ7473ζ7473ζ7572ζ767ζ767ζ7572    orthogonal lifted from D7
ρ82-2002-2ζ7473ζ7572ζ76774737677572ζ7572ζ767ζ767ζ7572ζ7473ζ74737473747375727677677572    orthogonal lifted from D14
ρ9220022ζ7572ζ767ζ7473ζ7572ζ7473ζ767ζ767ζ7473ζ7473ζ767ζ7572ζ7572ζ7572ζ7572ζ767ζ7473ζ7473ζ767    orthogonal lifted from D7
ρ10220022ζ767ζ7473ζ7572ζ767ζ7572ζ7473ζ7473ζ7572ζ7572ζ7473ζ767ζ767ζ767ζ767ζ7473ζ7572ζ7572ζ7473    orthogonal lifted from D7
ρ112-2002-2ζ7572ζ767ζ747375727473767ζ767ζ7473ζ7473ζ767ζ7572ζ75727572757276774737473767    orthogonal lifted from D14
ρ122-2002-2ζ767ζ7473ζ757276775727473ζ7473ζ7572ζ7572ζ7473ζ767ζ7677677677473757275727473    orthogonal lifted from D14
ρ132-200-11ζ767ζ7473ζ7572767757274733ζ743ζ73743ζ753ζ7275ζ3ζ753ζ727232ζ7432ζ737432ζ7632ζ776ζ32ζ7632ζ77ζ32ζ7632ζ77632ζ7632ζ7732ζ7432ζ73733ζ753ζ7272ζ3ζ753ζ72753ζ743ζ7373    orthogonal faithful
ρ142-200-11ζ767ζ7473ζ75727677572747332ζ7432ζ7374ζ3ζ753ζ72723ζ753ζ72753ζ743ζ7374ζ32ζ7632ζ7732ζ7632ζ77632ζ7632ζ77ζ32ζ7632ζ7763ζ743ζ7373ζ3ζ753ζ72753ζ753ζ727232ζ7432ζ7373    orthogonal faithful
ρ152-200-11ζ7473ζ7572ζ767747376775723ζ753ζ727532ζ7632ζ776ζ32ζ7632ζ77ζ3ζ753ζ727232ζ7432ζ73743ζ743ζ73743ζ743ζ737332ζ7432ζ7373ζ3ζ753ζ727532ζ7632ζ77ζ32ζ7632ζ7763ζ753ζ7272    orthogonal faithful
ρ162-200-11ζ7572ζ767ζ74737572747376732ζ7632ζ77632ζ7432ζ73743ζ743ζ7374ζ32ζ7632ζ77ζ3ζ753ζ72723ζ753ζ72753ζ753ζ7272ζ3ζ753ζ7275ζ32ζ7632ζ77632ζ7432ζ73733ζ743ζ737332ζ7632ζ77    orthogonal faithful
ρ172200-1-1ζ7572ζ767ζ7473ζ7572ζ7473ζ767ζ32ζ7632ζ773ζ743ζ737432ζ7432ζ737432ζ7632ζ7763ζ753ζ7275ζ3ζ753ζ72723ζ753ζ7275ζ3ζ753ζ7272ζ32ζ7632ζ7732ζ7432ζ73743ζ743ζ737432ζ7632ζ776    orthogonal lifted from D21
ρ182200-1-1ζ7473ζ7572ζ767ζ7473ζ767ζ75723ζ753ζ727532ζ7632ζ776ζ32ζ7632ζ77ζ3ζ753ζ727232ζ7432ζ73743ζ743ζ737432ζ7432ζ73743ζ743ζ73743ζ753ζ7275ζ32ζ7632ζ7732ζ7632ζ776ζ3ζ753ζ7272    orthogonal lifted from D21
ρ192200-1-1ζ7572ζ767ζ7473ζ7572ζ7473ζ76732ζ7632ζ77632ζ7432ζ73743ζ743ζ7374ζ32ζ7632ζ77ζ3ζ753ζ72723ζ753ζ7275ζ3ζ753ζ72723ζ753ζ727532ζ7632ζ7763ζ743ζ737432ζ7432ζ7374ζ32ζ7632ζ77    orthogonal lifted from D21
ρ202-200-11ζ7572ζ767ζ747375727473767ζ32ζ7632ζ773ζ743ζ737432ζ7432ζ737432ζ7632ζ7763ζ753ζ7275ζ3ζ753ζ7272ζ3ζ753ζ72753ζ753ζ727232ζ7632ζ773ζ743ζ737332ζ7432ζ7373ζ32ζ7632ζ776    orthogonal faithful
ρ212-200-11ζ7473ζ7572ζ76774737677572ζ3ζ753ζ7272ζ32ζ7632ζ7732ζ7632ζ7763ζ753ζ72753ζ743ζ737432ζ7432ζ737432ζ7432ζ73733ζ743ζ73733ζ753ζ7272ζ32ζ7632ζ77632ζ7632ζ77ζ3ζ753ζ7275    orthogonal faithful
ρ222200-1-1ζ767ζ7473ζ7572ζ767ζ7572ζ74733ζ743ζ73743ζ753ζ7275ζ3ζ753ζ727232ζ7432ζ737432ζ7632ζ776ζ32ζ7632ζ7732ζ7632ζ776ζ32ζ7632ζ773ζ743ζ7374ζ3ζ753ζ72723ζ753ζ727532ζ7432ζ7374    orthogonal lifted from D21
ρ232200-1-1ζ7473ζ7572ζ767ζ7473ζ767ζ7572ζ3ζ753ζ7272ζ32ζ7632ζ7732ζ7632ζ7763ζ753ζ72753ζ743ζ737432ζ7432ζ73743ζ743ζ737432ζ7432ζ7374ζ3ζ753ζ727232ζ7632ζ776ζ32ζ7632ζ773ζ753ζ7275    orthogonal lifted from D21
ρ242200-1-1ζ767ζ7473ζ7572ζ767ζ7572ζ747332ζ7432ζ7374ζ3ζ753ζ72723ζ753ζ72753ζ743ζ7374ζ32ζ7632ζ7732ζ7632ζ776ζ32ζ7632ζ7732ζ7632ζ77632ζ7432ζ73743ζ753ζ7275ζ3ζ753ζ72723ζ743ζ7374    orthogonal lifted from D21

Smallest permutation representation of D42
On 42 points
Generators in S42
(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)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)

G:=sub<Sym(42)| (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)>;

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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22) );

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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22)])

D42 is a maximal subgroup of   D21⋊C4  C3⋊D28  C7⋊D12  D84  C217D4  C2×S3×D7  Q8⋊D21
D42 is a maximal quotient of   Dic42  D84  C217D4

Matrix representation of D42 in GL2(𝔽41) generated by

201
400
,
01
10
G:=sub<GL(2,GF(41))| [20,40,1,0],[0,1,1,0] >;

D42 in GAP, Magma, Sage, TeX

D_{42}
% in TeX

G:=Group("D42");
// GroupNames label

G:=SmallGroup(84,14);
// by ID

G=gap.SmallGroup(84,14);
# by ID

G:=PCGroup([4,-2,-2,-3,-7,98,1155]);
// Polycyclic

G:=Group<a,b|a^42=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D42 in TeX
Character table of D42 in TeX

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