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G = Q8.6S4order 192 = 26·3

1st non-split extension by Q8 of S4 acting through Inn(Q8)

non-abelian, soluble

Aliases: Q8.6S4, 2+ 1+42S3, GL2(𝔽3)⋊7C22, CSU2(𝔽3)⋊4C22, SL2(𝔽3).6C23, C4.12(C2×S4), C4○D4.5D6, Q8.A42C2, C4.S44C2, C4.6S41C2, C2.17(C22×S4), C4.A4.5C22, Q8.7(C22×S3), SmallGroup(192,1483)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — Q8.6S4
C1C2Q8SL2(𝔽3)GL2(𝔽3)C4.6S4 — Q8.6S4
SL2(𝔽3) — Q8.6S4
C1C2Q8

Generators and relations for Q8.6S4
 G = < a,b,c,d,e,f | a4=e3=f2=1, b2=c2=d2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >

Subgroups: 487 in 140 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C4 [×4], C22 [×5], S3 [×2], C6, C8 [×4], C2×C4 [×9], D4 [×10], Q8 [×2], Q8 [×6], C23, Dic3 [×3], C12 [×3], D6, C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×3], C2×Q8 [×4], C4○D4 [×3], C4○D4 [×7], SL2(𝔽3), Dic6 [×3], C4×S3 [×3], C3×Q8, C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, CSU2(𝔽3) [×3], GL2(𝔽3), C4.A4 [×3], S3×Q8, D4○SD16, C4.S4 [×3], C4.6S4 [×3], Q8.A4, Q8.6S4
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, C2×S4 [×3], C22×S4, Q8.6S4

Character table of Q8.6S4

 class 12A2B2C2D2E34A4B4C4D4E4F4G68A8B8C8D8E12A12B12C
 size 116661282226121212866121212161616
ρ111111111111111111111111    trivial
ρ2111-1-1-111-1-11-1111-1-1-111-11-1    linear of order 2
ρ311-1-11-11-11-111-111-1-111-11-1-1    linear of order 2
ρ411-11-111-1-111-1-11111-11-1-1-11    linear of order 2
ρ5111-1-1111-1-111-1-11111-1-1-11-1    linear of order 2
ρ611111-111111-1-1-11-1-1-1-1-1111    linear of order 2
ρ711-11-1-11-1-11111-11-1-11-11-1-11    linear of order 2
ρ811-1-1111-11-11-11-1111-1-111-1-1    linear of order 2
ρ922-22-20-1-2-222000-10000011-1    orthogonal lifted from D6
ρ1022-2-220-1-22-22000-100000-111    orthogonal lifted from D6
ρ11222220-12222000-100000-1-1-1    orthogonal lifted from S3
ρ12222-2-20-12-2-22000-1000001-11    orthogonal lifted from D6
ρ133311-1-10-33-3-11-11011-1-11000    orthogonal lifted from C2×S4
ρ1433-111103-3-3-11-1-10-1-1-111000    orthogonal lifted from C2×S4
ρ1533-1-1-1-10333-1-1-1-1011111000    orthogonal lifted from S4
ρ16331-1110-3-33-1-1-110-1-11-11000    orthogonal lifted from C2×S4
ρ173311-110-33-3-1-11-10-1-111-1000    orthogonal lifted from C2×S4
ρ1833-111-103-3-3-1-1110111-1-1000    orthogonal lifted from C2×S4
ρ1933-1-1-110333-11110-1-1-1-1-1000    orthogonal lifted from S4
ρ20331-11-10-3-33-111-1011-11-1000    orthogonal lifted from C2×S4
ρ214-40000-2000000022-2-2-2000000    complex faithful
ρ224-40000-200000002-2-22-2000000    complex faithful
ρ238-8000020000000-200000000    symplectic faithful, Schur index 2

Smallest permutation representation of Q8.6S4
On 32 points
Generators in S32
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 24 3 22)(2 23 4 21)(5 27 7 25)(6 26 8 28)(9 20 11 18)(10 19 12 17)(13 29 15 31)(14 32 16 30)
(1 8 3 6)(2 5 4 7)(9 31 11 29)(10 32 12 30)(13 18 15 20)(14 19 16 17)(21 25 23 27)(22 26 24 28)
(1 18 3 20)(2 19 4 17)(5 14 7 16)(6 15 8 13)(9 22 11 24)(10 23 12 21)(25 30 27 32)(26 31 28 29)
(5 19 16)(6 20 13)(7 17 14)(8 18 15)(9 31 28)(10 32 25)(11 29 26)(12 30 27)
(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)(17 19)(18 20)(25 30)(26 31)(27 32)(28 29)

G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,24,3,22)(2,23,4,21)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30), (1,8,3,6)(2,5,4,7)(9,31,11,29)(10,32,12,30)(13,18,15,20)(14,19,16,17)(21,25,23,27)(22,26,24,28), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,22,11,24)(10,23,12,21)(25,30,27,32)(26,31,28,29), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,31,28)(10,32,25)(11,29,26)(12,30,27), (5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,19)(18,20)(25,30)(26,31)(27,32)(28,29)>;

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)(25,26,27,28)(29,30,31,32), (1,24,3,22)(2,23,4,21)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30), (1,8,3,6)(2,5,4,7)(9,31,11,29)(10,32,12,30)(13,18,15,20)(14,19,16,17)(21,25,23,27)(22,26,24,28), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,22,11,24)(10,23,12,21)(25,30,27,32)(26,31,28,29), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,31,28)(10,32,25)(11,29,26)(12,30,27), (5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,19)(18,20)(25,30)(26,31)(27,32)(28,29) );

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),(25,26,27,28),(29,30,31,32)], [(1,24,3,22),(2,23,4,21),(5,27,7,25),(6,26,8,28),(9,20,11,18),(10,19,12,17),(13,29,15,31),(14,32,16,30)], [(1,8,3,6),(2,5,4,7),(9,31,11,29),(10,32,12,30),(13,18,15,20),(14,19,16,17),(21,25,23,27),(22,26,24,28)], [(1,18,3,20),(2,19,4,17),(5,14,7,16),(6,15,8,13),(9,22,11,24),(10,23,12,21),(25,30,27,32),(26,31,28,29)], [(5,19,16),(6,20,13),(7,17,14),(8,18,15),(9,31,28),(10,32,25),(11,29,26),(12,30,27)], [(5,14),(6,15),(7,16),(8,13),(9,11),(10,12),(17,19),(18,20),(25,30),(26,31),(27,32),(28,29)])

Matrix representation of Q8.6S4 in GL4(𝔽3) generated by

0200
1000
0221
1211
,
2100
1100
0102
1110
,
2010
0012
1010
1110
,
2020
0121
2010
2122
,
2020
0121
1000
1021
,
1010
0212
0020
0011
G:=sub<GL(4,GF(3))| [0,1,0,1,2,0,2,2,0,0,2,1,0,0,1,1],[2,1,0,1,1,1,1,1,0,0,0,1,0,0,2,0],[2,0,1,1,0,0,0,1,1,1,1,1,0,2,0,0],[2,0,2,2,0,1,0,1,2,2,1,2,0,1,0,2],[2,0,1,1,0,1,0,0,2,2,0,2,0,1,0,1],[1,0,0,0,0,2,0,0,1,1,2,1,0,2,0,1] >;

Q8.6S4 in GAP, Magma, Sage, TeX

Q_8._6S_4
% in TeX

G:=Group("Q8.6S4");
// GroupNames label

G:=SmallGroup(192,1483);
// by ID

G=gap.SmallGroup(192,1483);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,1059,520,451,1684,655,172,1013,404,285,124]);
// 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=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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 Q8.6S4 in TeX

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