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G = Q8.5S4  order 192 = 26·3

3rd non-split extension by Q8 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: Q8.5S4, U2(𝔽3)⋊2C2, SL2(𝔽3)⋊4D4, 2+ 1+4⋊1S3, C4.6(C2×S4), C4○D4.2D6, Q8.A4⋊1C2, C4.3S4⋊2C2, Q8.5(C3⋊D4), C4.A4.2C22, C2.13(A4⋊D4), SmallGroup(192,988)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8 — C4.A4 — Q8.5S4
Chief series
C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — C4.3S4 — Q8.5S4
Lower central
SL2(𝔽3) — C4.A4 — Q8.5S4
Upper central
C1 — C2 — C4 — Q8

Generators and relations for Q8.5S4
 G = < a,b,c,d,e,f | a4=e3=f2=1, b2=c2=d2=a2, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=a-1b, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >

Subgroups: 405 in 78 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C23, C12, D6, C42, M4(2), D8, SD16, C2×D4, C4○D4, C4○D4, C3⋊C8, SL2(𝔽3), D12, C3×Q8, C4.D4, C4≀C2, C4⋊1D4, C8⋊C22, 2+ 1+4, Q8⋊2S3, GL2(𝔽3), C4.A4, C4.A4, D4⋊4D4, U2(𝔽3), C4.3S4, Q8.A4, Q8.5S4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, C2×S4, A4⋊D4, Q8.5S4

Character table of Q8.5S4

 class 12A2B2C2D34A4B4C4D4E68A8B12A12B12C
 size 11612248246121282424161616
ρ111111111111111111    trivial
ρ2111-1-111-11111-11-11-1    linear of order 2
ρ31111-11111-1-11-1-1111    linear of order 2
ρ4111-1111-11-1-111-1-11-1    linear of order 2
ρ5222-20-12-2200-1001-11    orthogonal lifted from D6
ρ622-2002-202002000-20    orthogonal lifted from D4
ρ722220-122200-100-1-1-1    orthogonal lifted from S3
ρ822-200-1-20200-100-√-31√-3    complex lifted from C3⋊D4
ρ922-200-1-20200-100√-31-√-3    complex lifted from C3⋊D4
ρ1033-1-11033-1-1-10-11000    orthogonal lifted from S4
ρ1133-11-103-3-1-1-1011000    orthogonal lifted from C2×S4
ρ1233-1-1-1033-11101-1000    orthogonal lifted from S4
ρ1333-11103-3-1110-1-1000    orthogonal lifted from C2×S4
ρ144-4000-20002-2200000    orthogonal faithful
ρ154-4000-2000-22200000    orthogonal faithful
ρ16662000-60-200000000    orthogonal lifted from A4⋊D4
ρ178-8000200000-200000    orthogonal faithful

Permutation representations of Q8.5S4
►On 24 points - transitive group 24T401
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 12 3 10)(2 11 4 9)(5 21 7 23)(6 24 8 22)(13 17 15 19)(14 20 16 18)

(1 12 3 10)(2 9 4 11)(5 8 7 6)(13 18 15 20)(14 19 16 17)(21 22 23 24)

(1 4 3 2)(5 24 7 22)(6 21 8 23)(9 10 11 12)(13 17 15 19)(14 18 16 20)

(1 17 7)(2 18 8)(3 19 5)(4 20 6)(9 16 24)(10 13 21)(11 14 22)(12 15 23)

(2 4)(5 19)(6 18)(7 17)(8 20)(9 12)(10 11)(13 22)(14 21)(15 24)(16 23)
 
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,12,3,10)(2,11,4,9)(5,21,7,23)(6,24,8,22)(13,17,15,19)(14,20,16,18), (1,12,3,10)(2,9,4,11)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,4,3,2)(5,24,7,22)(6,21,8,23)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,17,7)(2,18,8)(3,19,5)(4,20,6)(9,16,24)(10,13,21)(11,14,22)(12,15,23), (2,4)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,22)(14,21)(15,24)(16,23)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,12,3,10)(2,11,4,9)(5,21,7,23)(6,24,8,22)(13,17,15,19)(14,20,16,18), (1,12,3,10)(2,9,4,11)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,4,3,2)(5,24,7,22)(6,21,8,23)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,17,7)(2,18,8)(3,19,5)(4,20,6)(9,16,24)(10,13,21)(11,14,22)(12,15,23), (2,4)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,22)(14,21)(15,24)(16,23) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,12,3,10),(2,11,4,9),(5,21,7,23),(6,24,8,22),(13,17,15,19),(14,20,16,18)], [(1,12,3,10),(2,9,4,11),(5,8,7,6),(13,18,15,20),(14,19,16,17),(21,22,23,24)], [(1,4,3,2),(5,24,7,22),(6,21,8,23),(9,10,11,12),(13,17,15,19),(14,18,16,20)], [(1,17,7),(2,18,8),(3,19,5),(4,20,6),(9,16,24),(10,13,21),(11,14,22),(12,15,23)], [(2,4),(5,19),(6,18),(7,17),(8,20),(9,12),(10,11),(13,22),(14,21),(15,24),(16,23)]])
 
G:=TransitiveGroup(24,401);
 

Matrix representation of Q8.5S4 ►in GL4(ℚ) generated by

0001
0010
0-100
-1000
,
0010
000-1
-1000
0100
,
000-1
0010
0-100
1000
,
0-100
1000
0001
00-10
,
-1/21/2-1/21/2
-1/2-1/2-1/2-1/2
1/21/2-1/2-1/2
-1/21/21/2-1/2
,
-1/21/2-1/21/2
1/21/2-1/2-1/2
-1/2-1/2-1/2-1/2
1/2-1/2-1/21/2
G:=sub<GL(4,Rationals())| [0,0,0,-1,0,0,-1,0,0,1,0,0,1,0,0,0],[0,0,-1,0,0,0,0,1,1,0,0,0,0,-1,0,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[0,1,0,0,-1,0,0,0,0,0,0,-1,0,0,1,0],[-1/2,-1/2,1/2,-1/2,1/2,-1/2,1/2,1/2,-1/2,-1/2,-1/2,1/2,1/2,-1/2,-1/2,-1/2],[-1/2,1/2,-1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,1/2,-1/2,-1/2,1/2] >;
 

Q8.5S4 in GAP, Magma, Sage, TeX 

Q_8._5S_4
 
% in TeX
 
G:=Group("Q8.5S4");
 
// GroupNames label
 
G:=SmallGroup(192,988);
 
// by ID
 
G=gap.SmallGroup(192,988);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,680,2102,1059,520,451,1684,655,172,1013,404,285,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=e^3=f^2=1,b^2=c^2=d^2=a^2,b*a*b^-1=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^-1*b,d*c*d^-1=a^2*c,e*c*e^-1=a^2*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=a^2*d,f*e*f=e^-1>;
 
// generators/relations
 

Export

Character table of Q8.5S4 in TeX

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