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G = C24⋊Q8  order 128 = 27

The semidirect product of C24 and Q8 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C24⋊Q8, C24.3C23, (C22×C4)⋊Q8, C22⋊C4.6D4, C23⋊2Q8⋊3C2, C23.54(C2×Q8), C23.141(C2×D4), C22.43C22≀C2, C23.9D4⋊14C2, C22.4(C22⋊Q8), C23.130(C4○D4), C2.13(C23⋊Q8), C22.9(C4.4D4), (C22×D4).75C22, (C2×C23⋊C4).8C2, (C2×C22⋊C4).19C22, SmallGroup(128,764)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C24 — C24⋊Q8
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×C23⋊C4 — C24⋊Q8
Lower central
C1 — C2 — C24 — C24⋊Q8
Upper central
C1 — C2 — C24 — C24⋊Q8
Jennings
C1 — C2 — C24 — C24⋊Q8

Generators and relations for C24⋊Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, eae-1=acd, faf-1=abd, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 384 in 139 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C23⋊C4, C2×C22⋊C4, C22⋊Q8, C22×D4, C23.9D4, C2×C23⋊C4, C23⋊2Q8, C24⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22≀C2, C22⋊Q8, C4.4D4, C23⋊Q8, C24⋊Q8

Character table of C24⋊Q8

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M
 size 11222222288888888888888
ρ111111111111111111111111    trivial
ρ21111111111-1-1-11-1111-1-1-1-11    linear of order 2
ρ3111111111-1-11-1-11-11-11-11-11    linear of order 2
ρ4111111111-11-11-1-1-11-1-11-111    linear of order 2
ρ511111111111-1-1-111-1-111-1-1-1    linear of order 2
ρ61111111111-111-1-11-1-1-1-111-1    linear of order 2
ρ7111111111-1-1-1111-1-111-1-11-1    linear of order 2
ρ8111111111-111-11-1-1-11-111-1-1    linear of order 2
ρ922-2-22-22-2200000-200020000    orthogonal lifted from D4
ρ10222-2-222-2-200200000000-200    orthogonal lifted from D4
ρ1122-22-2-222-20000000-2000002    orthogonal lifted from D4
ρ1222-2-22-22-22000002000-20000    orthogonal lifted from D4
ρ1322-22-2-222-20000000200000-2    orthogonal lifted from D4
ρ14222-2-222-2-200-200000000200    orthogonal lifted from D4
ρ1522-2222-2-2-2-20000020000000    symplectic lifted from Q8, Schur index 2
ρ1622-2222-2-2-2200000-20000000    symplectic lifted from Q8, Schur index 2
ρ1722-2-2-22-222000-2i000000002i0    complex lifted from C4○D4
ρ182222-2-2-2-220000-2i0002i00000    complex lifted from C4○D4
ρ19222-22-2-22-20-2i000000002i000    complex lifted from C4○D4
ρ2022-2-2-22-2220002i00000000-2i0    complex lifted from C4○D4
ρ212222-2-2-2-2200002i000-2i00000    complex lifted from C4○D4
ρ22222-22-2-22-202i00000000-2i000    complex lifted from C4○D4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of C24⋊Q8
►On 16 points - transitive group 16T332
Generators in S16
(1 14)(2 10)(3 9)(4 15)(5 13)(6 11)(7 16)(8 12)

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

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

(1 7)(2 8)(3 6)(4 5)(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 4)(2 3)(5 7)(6 8)(9 14 11 16)(10 13 12 15)
 
G:=sub<Sym(16)| (1,14)(2,10)(3,9)(4,15)(5,13)(6,11)(7,16)(8,12), (1,2)(3,5)(4,6)(7,8)(9,13)(10,14)(11,15)(12,16), (1,6)(2,4)(3,7)(5,8)(9,16)(10,15)(11,14)(12,13), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,7)(6,8)(9,14,11,16)(10,13,12,15)>;
 
G:=Group( (1,14)(2,10)(3,9)(4,15)(5,13)(6,11)(7,16)(8,12), (1,2)(3,5)(4,6)(7,8)(9,13)(10,14)(11,15)(12,16), (1,6)(2,4)(3,7)(5,8)(9,16)(10,15)(11,14)(12,13), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,7)(6,8)(9,14,11,16)(10,13,12,15) );
 
G=PermutationGroup([[(1,14),(2,10),(3,9),(4,15),(5,13),(6,11),(7,16),(8,12)], [(1,2),(3,5),(4,6),(7,8),(9,13),(10,14),(11,15),(12,16)], [(1,6),(2,4),(3,7),(5,8),(9,16),(10,15),(11,14),(12,13)], [(1,7),(2,8),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,7),(6,8),(9,14,11,16),(10,13,12,15)]])
 
G:=TransitiveGroup(16,332);
 

►On 16 points - transitive group 16T354
Generators in S16
(2 10)(3 11)(6 14)(7 15)

(5 13)(6 14)(7 15)(8 16)

(2 10)(4 12)(6 14)(8 16)

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

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

(1 14 3 16)(2 13 4 15)(5 12 7 10)(6 11 8 9)
 
G:=sub<Sym(16)| (2,10)(3,11)(6,14)(7,15), (5,13)(6,14)(7,15)(8,16), (2,10)(4,12)(6,14)(8,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9)>;
 
G:=Group( (2,10)(3,11)(6,14)(7,15), (5,13)(6,14)(7,15)(8,16), (2,10)(4,12)(6,14)(8,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9) );
 
G=PermutationGroup([[(2,10),(3,11),(6,14),(7,15)], [(5,13),(6,14),(7,15),(8,16)], [(2,10),(4,12),(6,14),(8,16)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,3,16),(2,13,4,15),(5,12,7,10),(6,11,8,9)]])
 
G:=TransitiveGroup(16,354);
 

►On 16 points - transitive group 16T358
Generators in S16
(1 10)(2 9)(3 12)(4 11)(5 15)(6 14)(7 16)(8 13)

(1 3)(2 4)(9 11)(10 12)

(2 4)(6 7)(9 11)(14 16)

(1 3)(2 4)(5 8)(6 7)(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 7)(2 8)(3 6)(4 5)(9 13 11 15)(10 16 12 14)
 
G:=sub<Sym(16)| (1,10)(2,9)(3,12)(4,11)(5,15)(6,14)(7,16)(8,13), (1,3)(2,4)(9,11)(10,12), (2,4)(6,7)(9,11)(14,16), (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,7)(2,8)(3,6)(4,5)(9,13,11,15)(10,16,12,14)>;
 
G:=Group( (1,10)(2,9)(3,12)(4,11)(5,15)(6,14)(7,16)(8,13), (1,3)(2,4)(9,11)(10,12), (2,4)(6,7)(9,11)(14,16), (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,7)(2,8)(3,6)(4,5)(9,13,11,15)(10,16,12,14) );
 
G=PermutationGroup([[(1,10),(2,9),(3,12),(4,11),(5,15),(6,14),(7,16),(8,13)], [(1,3),(2,4),(9,11),(10,12)], [(2,4),(6,7),(9,11),(14,16)], [(1,3),(2,4),(5,8),(6,7),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,7),(2,8),(3,6),(4,5),(9,13,11,15),(10,16,12,14)]])
 
G:=TransitiveGroup(16,358);
 

►On 16 points - transitive group 16T368
Generators in S16
(1 16)(2 11)(3 9)(4 14)(5 12)(6 10)(7 15)(8 13)

(2 3)(5 6)(9 11)(10 12)

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

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

(5 6)(7 8)(9 10 11 12)(13 14 15 16)

(1 2)(3 4)(5 7)(6 8)(9 14 11 16)(10 13 12 15)
 
G:=sub<Sym(16)| (1,16)(2,11)(3,9)(4,14)(5,12)(6,10)(7,15)(8,13), (2,3)(5,6)(9,11)(10,12), (1,8)(2,5)(3,6)(4,7)(9,10)(11,12)(13,16)(14,15), (1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,2)(3,4)(5,7)(6,8)(9,14,11,16)(10,13,12,15)>;
 
G:=Group( (1,16)(2,11)(3,9)(4,14)(5,12)(6,10)(7,15)(8,13), (2,3)(5,6)(9,11)(10,12), (1,8)(2,5)(3,6)(4,7)(9,10)(11,12)(13,16)(14,15), (1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,2)(3,4)(5,7)(6,8)(9,14,11,16)(10,13,12,15) );
 
G=PermutationGroup([[(1,16),(2,11),(3,9),(4,14),(5,12),(6,10),(7,15),(8,13)], [(2,3),(5,6),(9,11),(10,12)], [(1,8),(2,5),(3,6),(4,7),(9,10),(11,12),(13,16),(14,15)], [(1,4),(2,3),(5,6),(7,8),(9,11),(10,12),(13,15),(14,16)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,2),(3,4),(5,7),(6,8),(9,14,11,16),(10,13,12,15)]])
 
G:=TransitiveGroup(16,368);
 

►On 16 points - transitive group 16T384
Generators in S16
(1 10)(2 9)(3 5)(4 8)(6 16)(7 15)(11 14)(12 13)

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

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

(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 10 3 12)(2 9 4 11)(5 15 7 13)(6 14 8 16)
 
G:=sub<Sym(16)| (1,10)(2,9)(3,5)(4,8)(6,16)(7,15)(11,14)(12,13), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15), (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), (1,10,3,12)(2,9,4,11)(5,15,7,13)(6,14,8,16)>;
 
G:=Group( (1,10)(2,9)(3,5)(4,8)(6,16)(7,15)(11,14)(12,13), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15), (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), (1,10,3,12)(2,9,4,11)(5,15,7,13)(6,14,8,16) );
 
G=PermutationGroup([[(1,10),(2,9),(3,5),(4,8),(6,16),(7,15),(11,14),(12,13)], [(1,3),(2,4),(5,10),(6,11),(7,12),(8,9),(13,15),(14,16)], [(1,3),(2,14),(4,16),(5,10),(6,8),(7,12),(9,11),(13,15)], [(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)], [(1,10,3,12),(2,9,4,11),(5,15,7,13),(6,14,8,16)]])
 
G:=TransitiveGroup(16,384);
 

Matrix representation of C24⋊Q8 ►in GL8(ℤ)

00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
0-1000000
00100000
000-10000
00000-100
00001000
0000000-1
00000010
,
10000000
0-1000000
00-100000
00010000
000000-10
00000001
00001000
00000-100

G:=sub<GL(8,Integers())| [0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0] >;
 

C24⋊Q8 in GAP, Magma, Sage, TeX 

C_2^4\rtimes Q_8
 
% in TeX
 
G:=Group("C2^4:Q8");
 
// GroupNames label
 
G:=SmallGroup(128,764);
 
// by ID
 
G=gap.SmallGroup(128,764);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,521,2804,4037,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,f*a*f^-1=a*b*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

Export

Character table of C24⋊Q8 in TeX

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