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G = C23⋊S4order 192 = 26·3

2nd semidirect product of C23 and S4 acting faithfully

non-abelian, soluble, monomial, rational

Aliases: C232S4, 2+ 1+44S3, C23⋊A42C2, C2.6(C22⋊S4), Aut(C22×C4), SmallGroup(192,1493)

Series: Derived Chief Lower central Upper central

C1C22+ 1+4C23⋊A4 — C23⋊S4
C1C2C232+ 1+4C23⋊A4 — C23⋊S4
C23⋊A4 — C23⋊S4
C1C2

Generators and relations for C23⋊S4
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=g2=1, ab=ba, eae=ac=ca, ad=da, faf-1=gag=b, dbd=ebe=bc=cb, fbf-1=abc, gbg=a, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 605 in 98 conjugacy classes, 8 normal (5 characteristic)
C1, C2, C2 [×5], C3, C4 [×4], C22 [×17], S3 [×2], C6, C2×C4 [×6], D4 [×9], Q8, C23 [×3], C23 [×6], A4 [×3], D6, C22⋊C4 [×6], C2×D4 [×6], C4○D4, C24, SL2(𝔽3), S4 [×6], C2×A4 [×3], C23⋊C4 [×3], C22≀C2 [×3], 2+ 1+4, C2×S4 [×3], C2≀C22, C23⋊A4, C23⋊S4
Quotients: C1, C2, S3, S4 [×3], C22⋊S4, C23⋊S4

Character table of C23⋊S4

 class 12A2B2C2D2E2F34A4B4C4D6
 size 116661212321224242432
ρ11111111111111    trivial
ρ211111-1-111-1-1-11    linear of order 2
ρ32222200-12000-1    orthogonal lifted from S3
ρ4333-1-1110-11-1-10    orthogonal lifted from S4
ρ533-13-1-1-10-11-110    orthogonal lifted from S4
ρ633-1-13110-1-1-110    orthogonal lifted from S4
ρ733-1-13-1-10-111-10    orthogonal lifted from S4
ρ833-13-1110-1-11-10    orthogonal lifted from S4
ρ9333-1-1-1-10-1-1110    orthogonal lifted from S4
ρ104-40002-210000-1    orthogonal faithful
ρ114-4000-2210000-1    orthogonal faithful
ρ1266-2-2-200020000    orthogonal lifted from C22⋊S4
ρ138-800000-100001    orthogonal faithful

Permutation representations of C23⋊S4
On 8 points - transitive group 8T39
Generators in S8
(1 4)(2 6)(3 7)(5 8)
(1 3)(2 8)(4 7)(5 6)
(1 2)(3 8)(4 6)(5 7)
(3 8)(5 7)
(3 8)(4 6)
(3 4 5)(6 7 8)
(1 2)(3 6)(4 8)(5 7)

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

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

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

G:=TransitiveGroup(8,39);

On 16 points - transitive group 16T442
Generators in S16
(1 13)(2 15)(3 8)(4 6)(5 9)(7 10)(11 14)(12 16)
(1 12)(2 14)(3 10)(4 5)(6 9)(7 8)(11 15)(13 16)
(1 2)(3 4)(5 10)(6 8)(7 9)(11 16)(12 14)(13 15)
(1 13)(2 15)(3 8)(4 6)(5 7)(9 10)(11 12)(14 16)
(1 11)(2 16)(3 9)(4 7)(5 6)(8 10)(12 13)(14 15)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 4)(2 3)(5 13)(6 12)(7 11)(8 14)(9 16)(10 15)

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

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

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

G:=TransitiveGroup(16,442);

On 24 points - transitive group 24T333
Generators in S24
(1 15)(2 16)(3 19)(4 22)(5 21)(6 17)(7 18)(8 13)(9 24)(10 20)(11 23)(12 14)
(1 18)(2 21)(3 14)(4 20)(5 16)(6 24)(7 15)(8 23)(9 17)(10 22)(11 13)(12 19)
(1 22)(2 23)(3 24)(4 15)(5 13)(6 14)(7 20)(8 21)(9 19)(10 18)(11 16)(12 17)
(1 15)(2 16)(3 19)(4 22)(5 8)(6 12)(7 10)(9 24)(11 23)(13 21)(14 17)(18 20)
(1 20)(2 13)(3 17)(4 10)(5 23)(6 9)(7 22)(8 11)(12 24)(14 19)(15 18)(16 21)
(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 22)(2 24)(3 23)(4 18)(5 17)(6 16)(7 20)(8 19)(9 21)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(24,333);

On 24 points - transitive group 24T431
Generators in S24
(1 7)(2 21)(4 10)(5 11)(6 14)(8 23)(12 17)(13 16)(15 18)(20 22)
(1 20)(3 9)(4 10)(5 13)(6 12)(7 22)(11 16)(14 17)(15 18)(19 24)
(1 22)(2 23)(3 24)(4 15)(5 13)(6 14)(7 20)(8 21)(9 19)(10 18)(11 16)(12 17)
(1 15)(2 13)(4 22)(5 23)(7 18)(8 11)(9 19)(10 20)(12 17)(16 21)
(2 13)(3 14)(5 23)(6 24)(7 20)(8 16)(9 12)(10 18)(11 21)(17 19)
(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 22)(2 24)(3 23)(4 15)(5 14)(6 13)(8 9)(10 18)(11 17)(12 16)(19 21)

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

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

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

G:=TransitiveGroup(24,431);

Polynomial with Galois group C23⋊S4 over ℚ
actionf(x)Disc(f)
8T39x8+x2+128·2292

Matrix representation of C23⋊S4 in GL4(ℤ) generated by

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

C23⋊S4 in GAP, Magma, Sage, TeX

C_2^3\rtimes S_4
% in TeX

G:=Group("C2^3:S4");
// GroupNames label

G:=SmallGroup(192,1493);
// by ID

G=gap.SmallGroup(192,1493);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,57,254,135,171,262,1684,1271,718,1013,516,530]);
// Polycyclic

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

Export

Character table of C23⋊S4 in TeX

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