Copied to
clipboard

G = Q8≀C2order 128 = 27

Wreath product of Q8 by C2

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

Aliases: Q8C2, C42.18D4, 2- 1+4.3C22, Q822C2, (C2×Q8).38D4, C2.29C2≀C22, C4⋊Q8.99C22, (C2×Q8).4C23, C42.3C43C2, D4.10D4.2C2, C22.53C22≀C2, C4.10D4.3C22, (C2×C4).22(C2×D4), 2-Sylow(Spin(5,3)), SmallGroup(128,937)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×Q8 — Q8≀C2
C1C2C22C2×C4C2×Q8C4⋊Q8Q82 — Q8≀C2
C1C2C22C2×Q8 — Q8≀C2
C1C2C22C2×Q8 — Q8≀C2
C1C2C22C2×Q8 — Q8≀C2

Generators and relations for Q8≀C2
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1b, cbc-1=a2b, bd=db, dcd=c3 >

Subgroups: 256 in 112 conjugacy classes, 28 normal (7 characteristic)
C1, C2, C2 [×2], C4 [×13], C22, C22, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4 [×4], Q8 [×12], C42 [×3], C42 [×3], C4⋊C4 [×9], M4(2) [×3], SD16 [×3], Q16 [×3], C2×Q8, C2×Q8 [×3], C2×Q8 [×3], C4○D4 [×4], C4.10D4 [×3], C4≀C2 [×3], C4×Q8 [×3], C4⋊Q8 [×3], C4⋊Q8 [×3], C8.C22 [×3], 2- 1+4, C42.3C4 [×3], D4.10D4 [×3], Q82, Q8≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, Q8≀C2

Character table of Q8≀C2

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C
 size 11284444444448888161616
ρ111111111111111111111    trivial
ρ2111-111111111111-11-1-1-1    linear of order 2
ρ3111-1-11-1-1111-111-1-1-11-11    linear of order 2
ρ41111-11-1-1111-111-11-1-11-1    linear of order 2
ρ5111-11-1-1111-1-11-1-1-11-111    linear of order 2
ρ611111-1-1111-1-11-1-1111-1-1    linear of order 2
ρ71111-1-11-111-111-111-1-1-11    linear of order 2
ρ8111-1-1-11-111-111-11-1-111-1    linear of order 2
ρ92220-200-2-2-20020002000    orthogonal lifted from D4
ρ1022202002-2-2002000-2000    orthogonal lifted from D4
ρ1122200020-2202-20-200000    orthogonal lifted from D4
ρ12222000-20-220-2-20200000    orthogonal lifted from D4
ρ1322200-2002-2-20-22000000    orthogonal lifted from D4
ρ14222002002-220-2-2000000    orthogonal lifted from D4
ρ1544-4200000000000-20000    orthogonal lifted from C2≀C22
ρ1644-4-20000000000020000    orthogonal lifted from C2≀C22
ρ174-400-2-2-22002200000000    symplectic faithful, Schur index 2
ρ184-400-222200-2-200000000    symplectic faithful, Schur index 2
ρ194-4002-22-2002-200000000    symplectic faithful, Schur index 2
ρ204-40022-2-200-2200000000    symplectic faithful, Schur index 2

Permutation representations of Q8≀C2
On 16 points - transitive group 16T370
Generators in S16
(1 5)(2 16 6 12)(3 7)(4 14 8 10)(9 13)(11 15)
(1 11 5 15)(2 12 6 16)(3 9 7 13)(4 10 8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 11)(3 14)(4 9)(5 12)(6 15)(7 10)(8 13)

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

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

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

G:=TransitiveGroup(16,370);

Matrix representation of Q8≀C2 in GL4(𝔽3) generated by

1200
2200
0010
0001
,
1200
2200
0012
0022
,
0012
0022
2200
2100
,
0010
0001
1000
0100
G:=sub<GL(4,GF(3))| [1,2,0,0,2,2,0,0,0,0,1,0,0,0,0,1],[1,2,0,0,2,2,0,0,0,0,1,2,0,0,2,2],[0,0,2,2,0,0,2,1,1,2,0,0,2,2,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

Q8≀C2 in GAP, Magma, Sage, TeX

Q_8\wr C_2
% in TeX

G:=Group("Q8wrC2");
// GroupNames label

G:=SmallGroup(128,937);
// by ID

G=gap.SmallGroup(128,937);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,723,352,2019,1018,297,248,2804,1971,718,375,172,4037]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

Export

Character table of Q8≀C2 in TeX

׿
×
𝔽