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G = C24.Q8  order 128 = 27

The non-split extension by C24 of Q8 acting faithfully

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

Aliases: C24.Q8, M4(2).14D4, C23.55(C2×Q8), C4.10C42⋊8C2, M4(2).C4⋊6C2, C4.157(C4⋊D4), C4.5(C42⋊2C2), (C22×C4).43C23, C22.7(C22⋊Q8), (C22×D4).91C22, (C2×M4(2)).29C22, C2.10(C23.Q8), (C2×C4).265(C2×D4), (C2×C4.D4).4C2, (C2×C4).352(C4○D4), SmallGroup(128,801)

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

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

Generators and relations for C24.Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=be2, 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=e3 >

Subgroups: 256 in 108 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, D4, C23, C23, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C24, C4.D4, C8.C4, C2×M4(2), C22×D4, C4.10C42, C2×C4.D4, M4(2).C4, C24.Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C42⋊2C2, C23.Q8, C24.Q8

Character table of C24.Q8

 class 12A2B2C2D2E2F4A4B4C4D8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222882222888888888888
ρ111111111111111111111111    trivial
ρ211111-1-11111-111-1-11-1-11-111    linear of order 2
ρ311111111111-11-11-111-1-1-1-1-1    linear of order 2
ρ411111-1-1111111-1-111-11-11-1-1    linear of order 2
ρ511111-1-11111-1-1111-111-1-11-1    linear of order 2
ρ6111111111111-11-1-1-1-1-1-111-1    linear of order 2
ρ711111-1-111111-1-11-1-11-111-11    linear of order 2
ρ811111111111-1-1-1-11-1-111-1-11    linear of order 2
ρ9222-2-2002-22-202000-2000000    orthogonal lifted from D4
ρ1022-2-22002-2-220020000000-20    orthogonal lifted from D4
ρ1122-2-22002-2-2200-2000000020    orthogonal lifted from D4
ρ1222-22-200-2-2220000-20020000    orthogonal lifted from D4
ρ1322-22-200-2-2220000200-20000    orthogonal lifted from D4
ρ14222-2-2002-22-20-20002000000    orthogonal lifted from D4
ρ1522222-22-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ16222222-2-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ1722-22-20022-2-200000000-2i002i    complex lifted from C4○D4
ρ1822-22-20022-2-2000000002i00-2i    complex lifted from C4○D4
ρ1922-2-2200-222-22i00000000-2i00    complex lifted from C4○D4
ρ20222-2-200-22-22000-2i002i00000    complex lifted from C4○D4
ρ21222-2-200-22-220002i00-2i00000    complex lifted from C4○D4
ρ2222-2-2200-222-2-2i000000002i00    complex lifted from C4○D4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of C24.Q8
►On 16 points - transitive group 16T369
Generators in S16
(1 7)(2 4)(3 5)(6 8)(9 11)(10 12)(13 15)(14 16)

(9 13)(10 14)(11 15)(12 16)

(2 6)(4 8)(9 13)(11 15)

(1 5)(2 6)(3 7)(4 8)(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 11 3 9 5 15 7 13)(2 14 4 12 6 10 8 16)
 
G:=sub<Sym(16)| (1,7)(2,4)(3,5)(6,8)(9,11)(10,12)(13,15)(14,16), (9,13)(10,14)(11,15)(12,16), (2,6)(4,8)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,3,9,5,15,7,13)(2,14,4,12,6,10,8,16)>;
 
G:=Group( (1,7)(2,4)(3,5)(6,8)(9,11)(10,12)(13,15)(14,16), (9,13)(10,14)(11,15)(12,16), (2,6)(4,8)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,3,9,5,15,7,13)(2,14,4,12,6,10,8,16) );
 
G=PermutationGroup([[(1,7),(2,4),(3,5),(6,8),(9,11),(10,12),(13,15),(14,16)], [(9,13),(10,14),(11,15),(12,16)], [(2,6),(4,8),(9,13),(11,15)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,11,3,9,5,15,7,13),(2,14,4,12,6,10,8,16)]])
 
G:=TransitiveGroup(16,369);
 

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

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

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

C24.Q8 in GAP, Magma, Sage, TeX 

C_2^4.Q_8
 
% in TeX
 
G:=Group("C2^4.Q8");
 
// GroupNames label
 
G:=SmallGroup(128,801);
 
// by ID
 
G=gap.SmallGroup(128,801);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,2019,521,248,2804,4037,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b*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^3>;
 
// generators/relations
 

Export

Character table of C24.Q8 in TeX

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