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G = C24.11Q8order 128 = 27

10th non-split extension by C24 of Q8 acting via Q8/C2=C22

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

Aliases: C24.11Q8, (C2×C8).7D4, C23.57(C2×Q8), (C22×C4).14Q8, C4.59(C41D4), C4.10C426C2, M4(2).C48C2, C24.4C4.1C2, (C23×C4).278C22, (C22×C4).742C23, C22.38(C22⋊Q8), C2.4(C23.4Q8), C4.120(C22.D4), (C2×M4(2)).238C22, (C2×C4).1384(C2×D4), (C2×C4).358(C4○D4), SmallGroup(128,823)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.11Q8
C1C2C22C23C22×C4C23×C4C24.4C4 — C24.11Q8
C1C2C22×C4 — C24.11Q8
C1C4C22×C4 — C24.11Q8
C1C2C2C22×C4 — C24.11Q8

Generators and relations for C24.11Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=be2, faf-1=ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=be3 >

Subgroups: 216 in 111 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4, C22 [×3], C22 [×11], C8 [×9], C2×C4 [×6], C2×C4 [×7], C23, C23 [×5], C2×C8 [×6], C2×C8 [×3], M4(2) [×12], C22×C4 [×2], C22×C4 [×6], C24, C22⋊C8 [×6], C8.C4 [×6], C2×M4(2) [×6], C23×C4, C4.10C42, C24.4C4 [×3], M4(2).C4 [×3], C24.11Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C23.4Q8, C24.11Q8

Character table of C24.11Q8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222441122244888888888888
ρ111111111111111111111111111    trivial
ρ211111-1-111111-1-111-11-111-11-1-1-1    linear of order 2
ρ311111-1-111111-1-1-111-1-11-11-111-1    linear of order 2
ρ411111111111111-11-1-111-1-1-1-1-11    linear of order 2
ρ511111-1-111111-1-1-1-11-11-11-11-111    linear of order 2
ρ611111111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ7111111111111111-111-1-1-1-1-1-11-1    linear of order 2
ρ811111-1-111111-1-11-1-111-1-11-11-11    linear of order 2
ρ9222-2-200222-2-20000002000000-2    orthogonal lifted from D4
ρ1022-2-220022-22-200000000020-200    orthogonal lifted from D4
ρ1122-2-220022-22-2000000000-20200    orthogonal lifted from D4
ρ12222-2-200222-2-2000000-20000002    orthogonal lifted from D4
ρ1322-22-20022-2-2200-200200000000    orthogonal lifted from D4
ρ1422-22-20022-2-2200200-200000000    orthogonal lifted from D4
ρ152222222-2-2-2-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ1622222-2-2-2-2-2-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ1722-22-200-2-222-200002i0000000-2i0    complex lifted from C4○D4
ρ1822-2-2200-2-22-2200000000-2i02i000    complex lifted from C4○D4
ρ1922-22-200-2-222-20000-2i00000002i0    complex lifted from C4○D4
ρ2022-2-2200-2-22-22000000002i0-2i000    complex lifted from C4○D4
ρ21222-2-200-2-2-222000-2i0002i000000    complex lifted from C4○D4
ρ22222-2-200-2-2-2220002i000-2i000000    complex lifted from C4○D4
ρ234-40002-24i-4i0002i-2i000000000000    complex faithful
ρ244-40002-2-4i4i000-2i2i000000000000    complex faithful
ρ254-4000-224i-4i000-2i2i000000000000    complex faithful
ρ264-4000-22-4i4i0002i-2i000000000000    complex faithful

Permutation representations of C24.11Q8
On 16 points - transitive group 16T403
Generators in S16
(10 14)(12 16)
(2 6)(4 8)(10 14)(12 16)
(9 13)(10 14)(11 15)(12 16)
(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 15 3 9 5 11 7 13)(2 14 8 12 6 10 4 16)

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

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

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

G:=TransitiveGroup(16,403);

Matrix representation of C24.11Q8 in GL4(𝔽5) generated by

4000
0100
0010
0001
,
4000
0400
0010
0001
,
4000
0100
0040
0001
,
4000
0400
0040
0004
,
0020
0003
4000
0100
,
0200
1000
0004
0020
G:=sub<GL(4,GF(5))| [4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,0,4,0,0,0,0,1,2,0,0,0,0,3,0,0],[0,1,0,0,2,0,0,0,0,0,0,2,0,0,4,0] >;

C24.11Q8 in GAP, Magma, Sage, TeX

C_2^4._{11}Q_8
% in TeX

G:=Group("C2^4.11Q8");
// GroupNames label

G:=SmallGroup(128,823);
// by ID

G=gap.SmallGroup(128,823);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,1018,248,1411,4037]);
// 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,f*a*f^-1=a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^3>;
// generators/relations

Export

Character table of C24.11Q8 in TeX

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