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G = Q8≀C2  order 128 = 27

Wreath product of Q8 by C2

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

Aliases: Q8≀C2, C42.18D4, 2- 1+4.3C22, Q82⋊2C2, (C2×Q8).38D4, C2.29C2≀C22, C4⋊Q8.99C22, (C2×Q8).4C23, C42.3C4⋊3C2, 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

Derived series
C1 — C2 — C2×Q8 — Q8≀C2
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C4⋊Q8 — Q82 — Q8≀C2
Lower central
C1 — C2 — C22 — C2×Q8 — Q8≀C2
Upper central
C1 — C2 — C22 — C2×Q8 — Q8≀C2
Jennings
C1 — C2 — C22 — C2×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, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C42, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4.10D4, C4≀C2, C4×Q8, C4⋊Q8, C4⋊Q8, C8.C22, 2- 1+4, C42.3C4, D4.10D4, Q82, Q8≀C2
Quotients: C1, C2, C22, D4, C23, C2×D4, 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 12 6 16)(3 7)(4 10 8 14)(9 13)(11 15)

(1 15 5 11)(2 16 6 12)(3 13 7 9)(4 14 8 10)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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