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G = Q8.S4order 192 = 26·3

2nd non-split extension by Q8 of S4 acting via S4/C22=S3

non-abelian, soluble, monomial

Aliases: Q8.2S4, C23.5S4, 2+ 1+4.3S3, C2.5(C22⋊S4), C23⋊A4.3C2, SmallGroup(192,1492)

Series: Derived Chief Lower central Upper central

C1C22+ 1+4C23⋊A4 — Q8.S4
C1C2Q82+ 1+4C23⋊A4 — Q8.S4
C23⋊A4 — Q8.S4
C1C2

Generators and relations for Q8.S4
 G = < a,b,c,d,e,f | a4=c2=d2=e3=1, b2=f2=a2, bab-1=cac=dad=fbf-1=a-1, eae-1=a-1b, faf-1=dbd=a2b, bc=cb, ebe-1=a, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >

Subgroups: 333 in 66 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2 [×2], C3, C4 [×4], C22 [×5], C6, C8 [×2], C2×C4 [×4], D4 [×4], Q8 [×2], Q8 [×2], C23, C23 [×2], Dic3, A4 [×2], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×2], SL2(𝔽3) [×2], C2×A4 [×2], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, CSU2(𝔽3) [×2], A4⋊C4, D4.9D4, C23⋊A4, Q8.S4
Quotients: C1, C2, S3, S4 [×3], C22⋊S4, Q8.S4

Character table of Q8.S4

 class 12A2B2C34A4B4C4D4E68A8B
 size 116123266121224322424
ρ11111111111111    trivial
ρ21111111-1-1-11-1-1    linear of order 2
ρ32222-122000-100    orthogonal lifted from S3
ρ4333-10-1-11110-1-1    orthogonal lifted from S4
ρ533-1-103-111-10-11    orthogonal lifted from S4
ρ6333-10-1-1-1-1-1011    orthogonal lifted from S4
ρ733-1-103-1-1-1101-1    orthogonal lifted from S4
ρ833-1-10-13-1-110-11    orthogonal lifted from S4
ρ933-1-10-1311-101-1    orthogonal lifted from S4
ρ104-4001002i-2i0-100    complex faithful
ρ114-400100-2i2i0-100    complex faithful
ρ1266-220-2-2000000    orthogonal lifted from C22⋊S4
ρ138-800-100000100    symplectic faithful, Schur index 2

Permutation representations of Q8.S4
On 16 points - transitive group 16T443
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 3 11)(2 12 4 10)(5 14 7 16)(6 13 8 15)
(1 4)(2 3)(5 13)(6 16)(7 15)(8 14)(9 10)(11 12)
(1 11)(2 10)(3 9)(4 12)(5 6)(7 8)(13 16)(14 15)
(2 9 10)(4 11 12)(5 6 13)(7 8 15)
(1 16 3 14)(2 7 4 5)(6 10 8 12)(9 15 11 13)

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

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

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

G:=TransitiveGroup(16,443);

On 16 points - transitive group 16T446
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 3 11)(2 12 4 10)(5 14 7 16)(6 13 8 15)
(2 4)(5 7)(10 12)(14 16)
(2 4)(6 8)(9 11)(14 16)
(2 9 10)(4 11 12)(5 14 8)(6 7 16)
(1 15 3 13)(2 8 4 6)(5 12 7 10)(9 14 11 16)

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

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

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

G:=TransitiveGroup(16,446);

Matrix representation of Q8.S4 in GL4(𝔽5) generated by

0010
0004
4000
0100
,
0002
0030
0300
2000
,
1000
0400
0040
0001
,
1000
0100
0040
0004
,
1000
0004
0300
0030
,
2000
0300
0001
0040
G:=sub<GL(4,GF(5))| [0,0,4,0,0,0,0,1,1,0,0,0,0,4,0,0],[0,0,0,2,0,0,3,0,0,3,0,0,2,0,0,0],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,0,3,0,0,0,0,3,0,4,0,0],[2,0,0,0,0,3,0,0,0,0,0,4,0,0,1,0] >;

Q8.S4 in GAP, Magma, Sage, TeX

Q_8.S_4
% in TeX

G:=Group("Q8.S4");
// GroupNames label

G:=SmallGroup(192,1492);
// by ID

G=gap.SmallGroup(192,1492);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,57,254,135,171,262,1684,1271,718,172,1013,2532,530,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^3=1,b^2=f^2=a^2,b*a*b^-1=c*a*c=d*a*d=f*b*f^-1=a^-1,e*a*e^-1=a^-1*b,f*a*f^-1=d*b*d=a^2*b,b*c=c*b,e*b*e^-1=a,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

Export

Character table of Q8.S4 in TeX

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