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G = Q82S4order 192 = 26·3

2nd semidirect product of Q8 and S4 acting via S4/C22=S3

non-abelian, soluble, monomial, rational

Aliases: Q82S4, C23.6S4, 2+ 1+45S3, C23⋊A43C2, C2.7(C22⋊S4), Hol(Q8), SmallGroup(192,1494)

Series: Derived Chief Lower central Upper central

C1C22+ 1+4C23⋊A4 — Q82S4
C1C2Q82+ 1+4C23⋊A4 — Q82S4
C23⋊A4 — Q82S4
C1C2

Generators and relations for Q82S4
 G = < a,b,c,d,e,f | a4=c2=d2=e3=f2=1, b2=a2, bab-1=cac=dad=fbf=a-1, eae-1=a-1b, faf=dbd=a2b, bc=cb, ebe-1=a, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 469 in 78 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22 [×8], S3 [×2], C6, C8 [×2], C2×C4 [×3], D4 [×10], Q8 [×2], C23, C23 [×3], A4 [×2], D6, C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4 [×5], C4○D4 [×2], SL2(𝔽3) [×2], S4 [×2], C2×A4 [×2], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, GL2(𝔽3) [×2], C2×S4, D44D4, C23⋊A4, Q82S4
Quotients: C1, C2, S3, S4 [×3], C22⋊S4, Q82S4

Character table of Q82S4

 class 12A2B2C2D34A4B4C4D68A8B
 size 116122432661212322424
ρ11111111111111    trivial
ρ21111-1111-1-11-1-1    linear of order 2
ρ322220-12200-100    orthogonal lifted from S3
ρ4333-1-10-1-1-1-1011    orthogonal lifted from S4
ρ533-1-1103-1-1-10-11    orthogonal lifted from S4
ρ633-1-1-10-13110-11    orthogonal lifted from S4
ρ733-1-110-13-1-101-1    orthogonal lifted from S4
ρ833-1-1-103-11101-1    orthogonal lifted from S4
ρ9333-110-1-1110-1-1    orthogonal lifted from S4
ρ104-4000100-22-100    orthogonal faithful
ρ114-40001002-2-100    orthogonal faithful
ρ1266-2200-2-200000    orthogonal lifted from C22⋊S4
ρ138-8000-10000100    orthogonal faithful

Permutation representations of Q82S4
On 8 points - transitive group 8T40
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 6 3 8)(2 5 4 7)
(2 4)(5 7)
(2 4)(6 8)
(2 6 7)(4 8 5)
(1 3)(2 6)(4 8)

G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,6,3,8)(2,5,4,7), (2,4)(5,7), (2,4)(6,8), (2,6,7)(4,8,5), (1,3)(2,6)(4,8)>;

G:=Group( (1,2,3,4)(5,6,7,8), (1,6,3,8)(2,5,4,7), (2,4)(5,7), (2,4)(6,8), (2,6,7)(4,8,5), (1,3)(2,6)(4,8) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8)], [(1,6,3,8),(2,5,4,7)], [(2,4),(5,7)], [(2,4),(6,8)], [(2,6,7),(4,8,5)], [(1,3),(2,6),(4,8)])

G:=TransitiveGroup(8,40);

On 16 points - transitive group 16T444
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 14)(2 5)(3 16)(4 7)(6 10)(8 12)(9 13)(11 15)

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

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

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

G:=TransitiveGroup(16,444);

On 16 points - transitive group 16T445
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 13)(2 6)(3 15)(4 8)(5 12)(7 10)(9 16)(11 14)

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

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

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

G:=TransitiveGroup(16,445);

On 24 points - transitive group 24T332
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 10 3 12)(2 9 4 11)(5 23 7 21)(6 22 8 24)(13 17 15 19)(14 20 16 18)
(2 4)(6 8)(9 11)(14 16)(18 20)(22 24)
(2 4)(5 7)(10 12)(13 15)(18 20)(22 24)
(1 17 23)(2 15 8)(3 19 21)(4 13 6)(5 9 20)(7 11 18)(10 16 24)(12 14 22)
(1 3)(2 10)(4 12)(5 20)(6 14)(7 18)(8 16)(13 22)(15 24)(17 21)(19 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), (1,10,3,12)(2,9,4,11)(5,23,7,21)(6,22,8,24)(13,17,15,19)(14,20,16,18), (2,4)(6,8)(9,11)(14,16)(18,20)(22,24), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24), (1,17,23)(2,15,8)(3,19,21)(4,13,6)(5,9,20)(7,11,18)(10,16,24)(12,14,22), (1,3)(2,10)(4,12)(5,20)(6,14)(7,18)(8,16)(13,22)(15,24)(17,21)(19,23)>;

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

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

G:=TransitiveGroup(24,332);

On 24 points - transitive group 24T430
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 7 3 5)(2 6 4 8)(9 14 11 16)(10 13 12 15)(17 23 19 21)(18 22 20 24)
(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(18 20)(22 24)
(1 3)(6 8)(9 12)(10 11)(13 14)(15 16)(17 21)(18 24)(19 23)(20 22)
(1 15 20)(2 10 21)(3 13 18)(4 12 23)(5 9 19)(6 14 22)(7 11 17)(8 16 24)
(2 5)(4 7)(6 8)(9 21)(10 19)(11 23)(12 17)(13 18)(14 24)(15 20)(16 22)

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

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

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

G:=TransitiveGroup(24,430);

Polynomial with Galois group Q82S4 over ℚ
actionf(x)Disc(f)
8T40x8+4x7-2x6-20x5-10x4+18x3+10x2-3x-133·174·593

Matrix representation of Q82S4 in GL4(ℤ) generated by

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

Q82S4 in GAP, Magma, Sage, TeX

Q_8\rtimes_2S_4
% in TeX

G:=Group("Q8:2S4");
// GroupNames label

G:=SmallGroup(192,1494);
// by ID

G=gap.SmallGroup(192,1494);
# by ID

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

G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^3=f^2=1,b^2=a^2,b*a*b^-1=c*a*c=d*a*d=f*b*f=a^-1,e*a*e^-1=a^-1*b,f*a*f=d*b*d=a^2*b,b*c=c*b,e*b*e^-1=a,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 Q82S4 in TeX

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