Copied to
clipboard

G = C4.3S4order 96 = 25·3

3rd non-split extension by C4 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: C4.3S4, Q8.5D6, GL2(F3):2C2, SL2(F3):2C22, C4.A4:1C2, C4oD4:2S3, C2.10(C2xS4), SmallGroup(96,193)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(F3) — C4.3S4
C1C2Q8SL2(F3)GL2(F3) — C4.3S4
SL2(F3) — C4.3S4
C1C2C4

Generators and relations for C4.3S4
 G = < a,b,c,d,e | a4=d3=e2=1, b2=c2=a2, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a2b, dbd-1=a2bc, ebe=bc, dcd-1=b, ece=a2c, ede=d-1 >

Subgroups: 173 in 42 conjugacy classes, 10 normal (8 characteristic)
Quotients: C1, C2, C22, S3, D6, S4, C2xS4, C4.3S4
6C2
12C2
12C2
4C3
3C22
3C4
6C22
6C22
12C22
12C22
4C6
8S3
8S3
3D4
3C8
3D4
3D4
3C8
3C2xC4
6C23
6D4
4D6
4D6
4C12
3D8
3M4(2)
3SD16
3D8
3SD16
3C2xD4
4D12
3C8:C22

Character table of C4.3S4

 class 12A2B2C2D34A4B68A8B12A12B
 size 11612128268121288
ρ11111111111111    trivial
ρ2111-1-11111-1-111    linear of order 2
ρ311-11-11-1111-1-1-1    linear of order 2
ρ411-1-111-111-11-1-1    linear of order 2
ρ522200-122-100-1-1    orthogonal lifted from S3
ρ622-200-1-22-10011    orthogonal lifted from D6
ρ733-11103-10-1-100    orthogonal lifted from S4
ρ833-1-1-103-101100    orthogonal lifted from S4
ρ9331-110-3-101-100    orthogonal lifted from C2xS4
ρ103311-10-3-10-1100    orthogonal lifted from C2xS4
ρ114-4000-20020000    orthogonal faithful
ρ124-4000100-100-33    orthogonal faithful
ρ134-4000100-1003-3    orthogonal faithful

Permutation representations of C4.3S4
On 16 points - transitive group 16T190
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 3 8)(2 7 4 5)(9 15 11 13)(10 16 12 14)
(1 9 3 11)(2 10 4 12)(5 16 7 14)(6 13 8 15)
(5 12 14)(6 9 15)(7 10 16)(8 11 13)
(1 4)(2 3)(5 13)(6 16)(7 15)(8 14)(9 10)(11 12)

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

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

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

G:=TransitiveGroup(16,190);

On 24 points - transitive group 24T127
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 12 3 10)(2 9 4 11)(5 8 7 6)(13 18 15 20)(14 19 16 17)(21 22 23 24)
(1 4 3 2)(5 24 7 22)(6 21 8 23)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 17 7)(2 18 8)(3 19 5)(4 20 6)(9 16 24)(10 13 21)(11 14 22)(12 15 23)
(2 4)(5 19)(6 18)(7 17)(8 20)(9 12)(10 11)(13 22)(14 21)(15 24)(16 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), (1,12,3,10)(2,9,4,11)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,4,3,2)(5,24,7,22)(6,21,8,23)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,17,7)(2,18,8)(3,19,5)(4,20,6)(9,16,24)(10,13,21)(11,14,22)(12,15,23), (2,4)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,22)(14,21)(15,24)(16,23)>;

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

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

G:=TransitiveGroup(24,127);

C4.3S4 is a maximal subgroup of
C8.4S4  C8.3S4  Q8.5S4  D4.3S4  GL2(F3):C22  Q8.7S4  D4.4S4  GL2(F3):S3  C12.7S4  C4.3S5  GL2(F3):D5  C20.3S4
C4.3S4 is a maximal quotient of
SL2(F3):Q8  Q8:D12  GL2(F3):C4  (C2xC4).S4  SL2(F3):D4  C12.4S4  GL2(F3):S3  C12.7S4  GL2(F3):D5  C20.3S4

Matrix representation of C4.3S4 in GL4(Q) generated by

000-1
00-10
0100
1000
,
000-1
0010
0-100
1000
,
0-100
1000
0001
00-10
,
-1/21/2-1/21/2
-1/2-1/2-1/2-1/2
1/21/2-1/2-1/2
-1/21/21/2-1/2
,
-1/21/2-1/21/2
1/21/2-1/2-1/2
-1/2-1/2-1/2-1/2
1/2-1/2-1/21/2
G:=sub<GL(4,Rationals())| [0,0,0,1,0,0,1,0,0,-1,0,0,-1,0,0,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[0,1,0,0,-1,0,0,0,0,0,0,-1,0,0,1,0],[-1/2,-1/2,1/2,-1/2,1/2,-1/2,1/2,1/2,-1/2,-1/2,-1/2,1/2,1/2,-1/2,-1/2,-1/2],[-1/2,1/2,-1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,1/2,-1/2,-1/2,1/2] >;

C4.3S4 in GAP, Magma, Sage, TeX

C_4._3S_4
% in TeX

G:=Group("C4.3S4");
// GroupNames label

G:=SmallGroup(96,193);
// by ID

G=gap.SmallGroup(96,193);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,601,295,146,579,447,117,364,286,202,88]);
// 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,e*a*e=a^-1,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 C4.3S4 in TeX
Character table of C4.3S4 in TeX

׿
x
:
Z
F
o
wr
Q
<