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G = Q83S4order 192 = 26·3

The semidirect product of Q8 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: Q83S4, A43SD16, A4⋊C83C2, C4⋊S4.2C2, C4.4(C2×S4), (Q8×A4)⋊1C2, (C2×A4).10D4, (C22×Q8)⋊1S3, (C22×C4).4D6, (C4×A4).4C22, C22⋊(Q82S3), C2.7(A4⋊D4), C23.20(C3⋊D4), SmallGroup(192,976)

Series: Derived Chief Lower central Upper central

C1C22C4×A4 — Q83S4
C1C22A4C2×A4C4×A4C4⋊S4 — Q83S4
A4C2×A4C4×A4 — Q83S4
C1C2C4Q8

Generators and relations for Q83S4
 G = < a,b,c,d,e,f | a4=c2=d2=e3=f2=1, b2=a2, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=a-1b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 354 in 77 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×4], C22, C22 [×5], S3, C6, C8 [×2], C2×C4 [×5], D4 [×4], Q8, Q8 [×4], C23, C23, C12 [×2], A4, D6, C22⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×3], C3⋊C8, D12, C3×Q8, S4, C2×A4, C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16 [×2], C22×Q8, Q82S3, C4×A4, C4×A4, C2×S4, Q8⋊D4, A4⋊C8, C4⋊S4, Q8×A4, Q83S4
Quotients: C1, C2 [×3], C22, S3, D4, D6, SD16, C3⋊D4, S4, Q82S3, C2×S4, A4⋊D4, Q83S4

Character table of Q83S4

 class 12A2B2C2D34A4B4C4D4E68A8B8C8D12A12B12C
 size 11332482461224812121212161616
ρ11111111111111111111    trivial
ρ21111-111-11-1-111111-1-11    linear of order 2
ρ31111111-11-111-1-1-1-1-1-11    linear of order 2
ρ41111-111111-11-1-1-1-1111    linear of order 2
ρ5222202-20-2002000000-2    orthogonal lifted from D4
ρ622220-122220-10000-1-1-1    orthogonal lifted from S3
ρ722220-12-22-20-1000011-1    orthogonal lifted from D6
ρ822220-1-20-200-10000--3-31    complex lifted from C3⋊D4
ρ922220-1-20-200-10000-3--31    complex lifted from C3⋊D4
ρ102-2-220200000-2-2--2--2-2000    complex lifted from SD16
ρ112-2-220200000-2--2-2-2--2000    complex lifted from SD16
ρ1233-1-1-103-3-1110-11-11000    orthogonal lifted from C2×S4
ρ1333-1-1-1033-1-1101-11-1000    orthogonal lifted from S4
ρ1433-1-11033-1-1-10-11-11000    orthogonal lifted from S4
ρ1533-1-1103-3-11-101-11-1000    orthogonal lifted from C2×S4
ρ164-4-440-20000020000000    orthogonal lifted from Q82S3
ρ1766-2-200-6020000000000    orthogonal lifted from A4⋊D4
ρ186-62-200000000--2--2-2-2000    complex faithful
ρ196-62-200000000-2-2--2--2000    complex faithful

Permutation representations of Q83S4
On 24 points - transitive group 24T326
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 23 3 21)(2 22 4 24)(5 16 7 14)(6 15 8 13)(9 19 11 17)(10 18 12 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 11 16)(2 12 13)(3 9 14)(4 10 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(2 4)(5 20)(6 19)(7 18)(8 17)(9 14)(10 13)(11 16)(12 15)(21 22)(23 24)

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

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

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

G:=TransitiveGroup(24,326);

Matrix representation of Q83S4 in GL5(𝔽73)

7212000
121000
00100
00010
00001
,
072000
10000
00100
00010
00001
,
10000
01000
000721
000720
001720
,
10000
01000
000172
001072
000072
,
10000
01000
007210
007200
007201
,
10000
6172000
00010
00100
00001

G:=sub<GL(5,GF(73))| [72,12,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,72,72,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,1,0,0,0,0,0,0,1],[1,61,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

Q83S4 in GAP, Magma, Sage, TeX

Q_8\rtimes_3S_4
% in TeX

G:=Group("Q8:3S4");
// GroupNames label

G:=SmallGroup(192,976);
// by ID

G=gap.SmallGroup(192,976);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,85,64,254,135,58,1124,4037,285,2358,475]);
// Polycyclic

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

Export

Character table of Q83S4 in TeX

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