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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.A41C2, C4○D42S3, C2.10(C2×S4), 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 >

6C2
12C2
12C2
4C3
3C22
3C4
6C22
6C22
12C22
12C22
4C6
8S3
8S3
3D4
3C8
3D4
3D4
3C8
3C2×C4
6C23
6D4
4D6
4D6
4C12
3D8
3M4(2)
3SD16
3D8
3SD16
3C2×D4
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 C2×S4
ρ103311-10-3-10-1100    orthogonal lifted from C2×S4
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 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 4)(2 3)(5 15)(6 14)(7 13)(8 16)(9 10)(11 12)

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

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

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)], [(1,4),(2,3),(5,15),(6,14),(7,13),(8,16),(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  (C2×C4).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

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