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G = C23.9S4order 192 = 26·3

3rd non-split extension by C23 of S4 acting via S4/C22=S3

non-abelian, soluble, monomial

Aliases: C23.9S4, C422Dic3, C42⋊C33C4, (C2×C42).S3, C2.1(C42⋊S3), C22.1(A4⋊C4), (C2×C42⋊C3).3C2, SmallGroup(192,182)

Series: Derived Chief Lower central Upper central

C1C42C42⋊C3 — C23.9S4
C1C22C42C42⋊C3C2×C42⋊C3 — C23.9S4
C42⋊C3 — C23.9S4
C1C2

Generators and relations for C23.9S4
 G = < a,b,c,d,e,f,g | a2=b2=c2=f3=1, d2=cb=gbg-1=fcf-1=bc, e2=fbf-1=c, g2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bd=db, be=eb, fef-1=cd=dc, geg-1=ce=ec, cg=gc, fdf-1=gdg-1=de=ed, gfg-1=f-1 >

3C2
3C2
16C3
3C22
3C4
3C4
3C22
3C4
3C4
12C4
12C4
16C6
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C2×C4
6C2×C4
6C8
6C2×C4
6C8
4A4
16Dic3
3C22×C4
3C42
3C4⋊C4
3C4⋊C4
3M4(2)
3M4(2)
6C42
6C2×C8
6M4(2)
6C22⋊C4
4C2×A4
3C2×M4(2)
3C42⋊C2
4A4⋊C4
3C426C4

Character table of C23.9S4

 class 12A2B2C34A4B4C4D4E4F4G4H4I4J68A8B8C8D
 size 113332333366121212123212121212
ρ111111111111111111111    trivial
ρ211111111111-1-1-1-11-1-1-1-1    linear of order 2
ρ31-11-11-11-11-11i-ii-i-1i-ii-i    linear of order 4
ρ41-11-11-11-11-11-ii-ii-1-ii-ii    linear of order 4
ρ52222-12222220000-10000    orthogonal lifted from S3
ρ62-22-2-1-22-22-22000010000    symplectic lifted from Dic3, Schur index 2
ρ733330-1-1-1-1-1-111110-1-1-1-1    orthogonal lifted from S4
ρ833330-1-1-1-1-1-1-1-1-1-101111    orthogonal lifted from S4
ρ93-3-1101+2i-1+2i1-2i-1-2i-11i-i-ii0-111-1    complex faithful
ρ1033-1-10-1-2i-1+2i-1+2i-1-2i1111-1-10ii-i-i    complex lifted from C42⋊S3
ρ113-3-1101+2i-1+2i1-2i-1-2i-11-iii-i01-1-11    complex faithful
ρ1233-1-10-1+2i-1-2i-1-2i-1+2i1111-1-10-i-iii    complex lifted from C42⋊S3
ρ133-33-301-11-11-1-ii-ii0i-ii-i    complex lifted from A4⋊C4
ρ143-3-1101-2i-1-2i1+2i-1+2i-11i-i-ii01-1-11    complex faithful
ρ1533-1-10-1+2i-1-2i-1-2i-1+2i11-1-1110ii-i-i    complex lifted from C42⋊S3
ρ163-33-301-11-11-1i-ii-i0-ii-ii    complex lifted from A4⋊C4
ρ173-3-1101-2i-1-2i1+2i-1+2i-11-iii-i0-111-1    complex faithful
ρ1833-1-10-1-2i-1+2i-1+2i-1-2i11-1-1110-i-iii    complex lifted from C42⋊S3
ρ1966-2-202222-2-2000000000    orthogonal lifted from C42⋊S3
ρ206-6-220-22-222-2000000000    symplectic faithful, Schur index 2

Permutation representations of C23.9S4
On 12 points - transitive group 12T98
Generators in S12
(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)
(1 2)(3 4)(9 11)(10 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3 2 4)(5 6 7 8)
(1 9 8)(2 11 6)(3 12 7)(4 10 5)
(1 5 2 7)(3 8 4 6)(9 10 11 12)

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

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

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

G:=TransitiveGroup(12,98);

On 24 points - transitive group 24T317
Generators in S24
(1 2)(3 4)(5 8)(6 7)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 8)(6 7)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 8)(6 7)(9 11)(10 12)(13 15)(14 16)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3 2 4)(5 7 8 6)(9 12 11 10)(13 16 15 14)(17 19)(18 20)(21 23)(22 24)
(1 23 9)(2 21 11)(3 24 12)(4 22 10)(5 19 14)(6 18 15)(7 20 13)(8 17 16)
(1 16 2 14)(3 13 4 15)(5 9 8 11)(6 12 7 10)(17 21 19 23)(18 24 20 22)

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

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

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

G:=TransitiveGroup(24,317);

On 24 points - transitive group 24T481
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(17 19)(18 20)(21 23)(22 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 4 2 3)(5 7 6 8)(17 18 19 20)(21 22 23 24)
(1 12 20)(2 10 18)(3 9 17)(4 11 19)(5 13 21)(6 15 23)(7 16 24)(8 14 22)
(1 23 2 21)(3 24 4 22)(5 20 6 18)(7 19 8 17)(9 16 11 14)(10 13 12 15)

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

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

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

G:=TransitiveGroup(24,481);

On 24 points - transitive group 24T482
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(21 23)(22 24)
(1 5)(2 6)(3 7)(4 8)(13 15)(14 16)(17 19)(18 20)
(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 3 5 7)(2 4 6 8)(9 22)(10 23)(11 24)(12 21)(13 20 15 18)(14 17 16 19)
(1 9 14)(2 24 18)(3 21 13)(4 10 17)(5 11 16)(6 22 20)(7 23 15)(8 12 19)
(1 13 2 17)(3 18 4 14)(5 15 6 19)(7 20 8 16)(9 21 24 10)(11 23 22 12)

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

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

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

G:=TransitiveGroup(24,482);

Polynomial with Galois group C23.9S4 over ℚ
actionf(x)Disc(f)
12T98x12-14210x10-5541749x8+46928284x6-72912385x4-52237562x2+8560357268·520·1314·3713·140837291602386138174

Matrix representation of C23.9S4 in GL3(𝔽5) generated by

400
040
004
,
224
040
323
,
242
332
004
,
122
124
142
,
241
104
414
,
101
004
014
,
202
023
003
G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[2,0,3,2,4,2,4,0,3],[2,3,0,4,3,0,2,2,4],[1,1,1,2,2,4,2,4,2],[2,1,4,4,0,1,1,4,4],[1,0,0,0,0,1,1,4,4],[2,0,0,0,2,0,2,3,3] >;

C23.9S4 in GAP, Magma, Sage, TeX

C_2^3._9S_4
% in TeX

G:=Group("C2^3.9S4");
// GroupNames label

G:=SmallGroup(192,182);
// by ID

G=gap.SmallGroup(192,182);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,14,170,675,185,360,424,1173,102,6053,1027,1784]);
// Polycyclic

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

Export

Subgroup lattice of C23.9S4 in TeX
Character table of C23.9S4 in TeX

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