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G = D4≀C2order 128 = 27

Wreath product of D4 by C2

p-group, non-abelian, nilpotent (class 4), monomial, rational

Aliases: D4C2, C2D4, C242D4, C423D4, 2+ 1+41C22, D422C2, C2≀C41C2, (C2×D4)⋊3D4, C22⋊C41D4, C23⋊C4⋊C22, C2≀C221C2, D44D41C2, C42⋊C45C2, C2.20C2≀C22, (C2×D4).1C23, C23.13(C2×D4), C4.D41C22, C22≀C2.3C22, C22.44C22≀C2, C41D4.55C22, (C2×C4).13(C2×D4), 2-Sylow(S8), SmallGroup(128,928)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×D4 — D4≀C2
C1C2C22C23C2×D4C22≀C2D42 — D4≀C2
C1C2C22C2×D4 — D4≀C2
C1C2C22C2×D4 — D4≀C2
C1C2C22C2×D4 — D4≀C2

Generators and relations for D4≀C2
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, eae-1=faf=ab=ba, ac=ca, ad=da, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >

Subgroups: 576 in 177 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×9], C4 [×8], C22, C22 [×28], C8, C2×C4, C2×C4 [×12], D4 [×26], Q8, C23 [×2], C23 [×16], C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4, M4(2), D8, SD16, C22×C4 [×2], C2×D4 [×2], C2×D4 [×20], C4○D4 [×2], C24 [×2], C24, C23⋊C4 [×2], C23⋊C4 [×2], C4.D4, C4≀C2, C4×D4, C22≀C2 [×2], C22≀C2 [×3], C4⋊D4 [×2], C41D4, C8⋊C22, C22×D4 [×2], 2+ 1+4, C2≀C4 [×2], C42⋊C4, D44D4, C2≀C22 [×2], D42, D4≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, D4≀C2

Character table of D4≀C2

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H8
 size 11244444488444888161616
ρ111111111111111111111    trivial
ρ2111-111-111-11-1-11-111-11-1    linear of order 2
ρ3111-11-1-1-1111111-11-1-1-11    linear of order 2
ρ411111-11-11-11-1-1111-11-1-1    linear of order 2
ρ511111-11-11-1-1-1-111-1-1-111    linear of order 2
ρ6111-11-1-1-111-1111-1-1-111-1    linear of order 2
ρ7111-111-111-1-1-1-11-1-111-11    linear of order 2
ρ81111111111-11111-11-1-1-1    linear of order 2
ρ92220-220220000-200-2000    orthogonal lifted from D4
ρ102220-2000-2-20222000000    orthogonal lifted from D4
ρ11222-220-20-20000-2200000    orthogonal lifted from D4
ρ122220-2000-220-2-22000000    orthogonal lifted from D4
ρ132220-2-20-220000-2002000    orthogonal lifted from D4
ρ1422222020-20000-2-200000    orthogonal lifted from D4
ρ154-4020-2-22000-220000000    orthogonal faithful
ρ1644-4000000020000-20000    orthogonal lifted from C2≀C22
ρ174-40202-2-20002-20000000    orthogonal faithful
ρ184-40-2022-2000-220000000    orthogonal faithful
ρ1944-40000000-2000020000    orthogonal lifted from C2≀C22
ρ204-40-20-2220002-20000000    orthogonal faithful

Permutation representations of D4≀C2
On 8 points - transitive group 8T35
Generators in S8
(1 5)(2 3)(4 7)(6 8)
(1 7)(2 6)(3 8)(4 5)
(2 3)(6 8)
(1 4)(2 3)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)
(1 3)(2 4)(5 6)(7 8)

G:=sub<Sym(8)| (1,5)(2,3)(4,7)(6,8), (1,7)(2,6)(3,8)(4,5), (2,3)(6,8), (1,4)(2,3)(5,7)(6,8), (1,2)(3,4)(5,6,7,8), (1,3)(2,4)(5,6)(7,8)>;

G:=Group( (1,5)(2,3)(4,7)(6,8), (1,7)(2,6)(3,8)(4,5), (2,3)(6,8), (1,4)(2,3)(5,7)(6,8), (1,2)(3,4)(5,6,7,8), (1,3)(2,4)(5,6)(7,8) );

G=PermutationGroup([(1,5),(2,3),(4,7),(6,8)], [(1,7),(2,6),(3,8),(4,5)], [(2,3),(6,8)], [(1,4),(2,3),(5,7),(6,8)], [(1,2),(3,4),(5,6,7,8)], [(1,3),(2,4),(5,6),(7,8)])

G:=TransitiveGroup(8,35);

On 16 points - transitive group 16T376
Generators in S16
(1 11)(2 16)(3 7)(4 14)(5 15)(6 12)(8 10)(9 13)
(1 5)(2 12)(3 9)(4 8)(6 16)(7 13)(10 14)(11 15)
(2 16)(4 14)(6 12)(8 10)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,376);

On 16 points - transitive group 16T388
Generators in S16
(1 11)(2 7)(3 13)(4 5)(6 15)(8 9)(10 14)(12 16)
(1 15)(2 12)(3 9)(4 14)(5 10)(6 11)(7 16)(8 13)
(2 7)(4 5)(10 14)(12 16)
(1 6)(2 7)(3 8)(4 5)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 8)(3 7)(4 6)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,388);

On 16 points - transitive group 16T390
Generators in S16
(2 11)(3 13)(4 9)(5 12)(6 14)(8 16)
(1 10)(2 8)(3 5)(4 9)(6 14)(7 15)(11 16)(12 13)
(2 16)(4 14)(6 9)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 16)(14 15)

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

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

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

G:=TransitiveGroup(16,390);

On 16 points - transitive group 16T391
Generators in S16
(1 3)(2 9)(4 6)(5 12)(7 10)(8 16)(11 14)(13 15)
(1 5)(2 6)(3 12)(4 9)(7 13)(8 14)(10 15)(11 16)
(1 15)(3 13)(5 10)(7 12)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 16)(14 15)

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

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

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

G:=TransitiveGroup(16,391);

On 16 points - transitive group 16T393
Generators in S16
(1 6)(2 7)(3 9)(4 5)(8 16)(10 13)(11 14)(12 15)
(2 15)(3 16)(7 12)(8 9)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)

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

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

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

G:=TransitiveGroup(16,393);

On 16 points - transitive group 16T395
Generators in S16
(3 16)(4 9)(5 12)(6 13)
(2 15)(3 5)(4 9)(6 13)(8 11)(12 16)
(1 10)(2 15)(3 12)(4 13)(5 16)(6 9)(7 14)(8 11)
(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(6 8)(9 15)(10 14)(11 13)(12 16)

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

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

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

G:=TransitiveGroup(16,395);

On 16 points - transitive group 16T396
Generators in S16
(3 16)(8 9)
(2 15)(3 16)(7 12)(8 9)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)

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

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

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

G:=TransitiveGroup(16,396);

On 16 points - transitive group 16T401
Generators in S16
(2 12)(4 16)(6 10)(8 14)
(1 11)(2 12)(3 15)(4 16)(5 9)(6 10)(7 13)(8 14)
(1 5)(3 7)(9 11)(13 15)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,401);

Polynomial with Galois group D4≀C2 over ℚ
actionf(x)Disc(f)
8T35x8-x7-x6-x5-x3-x2-x+1-28·54·892

Matrix representation of D4≀C2 in GL4(ℤ) generated by

-1000
0100
00-10
000-1
,
1000
0-100
0010
000-1
,
1000
0100
00-10
000-1
,
-1000
0-100
00-10
000-1
,
00-10
000-1
0-100
-1000
,
0010
0001
1000
0100
G:=sub<GL(4,Integers())| [-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,0,-1,0,0,-1,0,-1,0,0,0,0,-1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

D4≀C2 in GAP, Magma, Sage, TeX

D_4\wr C_2
% in TeX

G:=Group("D4wrC2");
// GroupNames label

G:=SmallGroup(128,928);
// by ID

G=gap.SmallGroup(128,928);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,297,1971,375,4037]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,e*a*e^-1=f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

Export

Character table of D4≀C2 in TeX

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