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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, C4GL2(𝔽3), C4CSU2(𝔽3), GL2(𝔽3)⋊3C2, CSU2(𝔽3)⋊3C2, SL2(𝔽3).4C22, C4.A42C2, C4○D41S3, C2.9(C2×S4), 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 >

6C2
12C2
4C3
3C4
3C22
6C4
6C22
4C6
4S3
4S3
3C8
3D4
3Q8
3C2×C4
3D4
3C8
6D4
6C2×C4
4D6
4C12
4Dic3
3Q16
3SD16
3C2×C8
3C4○D4
3SD16
3D8
4C4×S3
3C4○D8

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 C2×S4
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 C2×S4
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
C4×CSU2(𝔽3)  Q8.Dic6  C4×GL2(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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