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G = C4.6S4order 96 = 25·3

3rd central extension by C4 of S4

non-abelian, soluble

Aliases: C4.6S4, Q8.4D6, C4oGL2(F3), C4oCSU2(F3), GL2(F3):3C2, CSU2(F3):3C2, SL2(F3).4C22, C4.A4:2C2, C4oD4:1S3, C2.9(C2xS4), SmallGroup(96,192)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(F3) — C4.6S4
C1C2Q8SL2(F3)GL2(F3) — C4.6S4
SL2(F3) — C4.6S4
C1C4

Generators and relations for C4.6S4
 G = < a,b,c,d,e | a4=d3=e2=1, b2=c2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a2b, dbd-1=a2bc, ebe=bc, dcd-1=b, ece=a2c, ede=d-1 >

Subgroups: 129 in 39 conjugacy classes, 10 normal (all characteristic)
Quotients: C1, C2, C22, S3, D6, S4, C2xS4, C4.6S4
6C2
12C2
4C3
3C4
3C22
6C4
6C22
4C6
4S3
4S3
3C8
3D4
3Q8
3C2xC4
3D4
3C8
6D4
6C2xC4
4D6
4C12
4Dic3
3Q16
3SD16
3C2xC8
3C4oD4
3SD16
3D8
4C4xS3
3C4oD8

Character table of C4.6S4

 class 12A2B2C34A4B4C4D68A8B8C8D12A12B
 size 116128116128666688
ρ11111111111111111    trivial
ρ2111-11111-11-1-1-1-111    linear of order 2
ρ311-1-11-1-11111-11-1-1-1    linear of order 2
ρ411-111-1-11-11-11-11-1-1    linear of order 2
ρ52220-12220-10000-1-1    orthogonal lifted from S3
ρ622-20-1-2-220-1000011    orthogonal lifted from D6
ρ72-200-12i-2i001-2--22-2-ii    complex faithful
ρ82-200-1-2i2i001-2-22--2i-i    complex faithful
ρ92-200-12i-2i0012-2-2--2-ii    complex faithful
ρ102-200-1-2i2i0012--2-2-2i-i    complex faithful
ρ11331-10-3-3-110-11-1100    orthogonal lifted from C2xS4
ρ1233-11033-110-1-1-1-100    orthogonal lifted from S4
ρ1333-1-1033-1-10111100    orthogonal lifted from S4
ρ1433110-3-3-1-101-11-100    orthogonal lifted from C2xS4
ρ154-4001-4i4i00-10000-ii    complex faithful
ρ164-40014i-4i00-10000i-i    complex faithful

Permutation representations of C4.6S4
On 16 points - transitive group 16T189
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8 3 6)(2 5 4 7)(9 15 11 13)(10 16 12 14)
(1 9 3 11)(2 10 4 12)(5 14 7 16)(6 15 8 13)
(5 10 16)(6 11 13)(7 12 14)(8 9 15)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)

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

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

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

G:=TransitiveGroup(16,189);

C4.6S4 is a maximal subgroup of
CU2(F3)  C8.5S4  GL2(F3):C22  Q8.6S4  Q8.7S4  D4.4S4  D4.5S4  Dic3.4S4  Dic3.5S4  C12.14S4  C4.6S5  Dic5.6S4  Dic5.7S4  C20.6S4
C4.6S4 is a maximal quotient of
C4xCSU2(F3)  Q8.Dic6  C4xGL2(F3)  Q8.2D12  C4.A4:C4  SL2(F3).D4  SL2(F3):D4  C12.11S4  Dic3.4S4  Dic3.5S4  C12.14S4  Dic5.6S4  Dic5.7S4  C20.6S4

Matrix representation of C4.6S4 in GL2(F17) generated by

40
04
,
99
68
,
611
911
,
110
1515
,
110
016
G:=sub<GL(2,GF(17))| [4,0,0,4],[9,6,9,8],[6,9,11,11],[1,15,10,15],[1,0,10,16] >;

C4.6S4 in GAP, Magma, Sage, TeX

C_4._6S_4
% in TeX

G:=Group("C4.6S4");
// GroupNames label

G:=SmallGroup(96,192);
// by ID

G=gap.SmallGroup(96,192);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,295,146,579,447,117,364,286,202,88]);
// Polycyclic

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

Export

Subgroup lattice of C4.6S4 in TeX
Character table of C4.6S4 in TeX

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