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G = C6.6S4order 144 = 24·32

6th non-split extension by C6 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: C6.6S4, C3⋊GL2(𝔽3), SL2(𝔽3)⋊S3, Q8⋊(C3⋊S3), (C3×Q8)⋊1S3, C2.3(C3⋊S4), (C3×SL2(𝔽3))⋊1C2, SmallGroup(144,125)

Series: Derived Chief Lower central Upper central

C1C2Q8C3×SL2(𝔽3) — C6.6S4
C1C2Q8C3×Q8C3×SL2(𝔽3) — C6.6S4
C3×SL2(𝔽3) — C6.6S4
C1C2

Generators and relations for C6.6S4
 G = < a,b,c,d,e | a6=d3=e2=1, b2=c2=a3, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a3b, dbd-1=a3bc, ebe=bc, dcd-1=b, ece=a3c, ede=d-1 >

36C2
4C3
4C3
4C3
3C4
18C22
4C6
4C6
4C6
12S3
12S3
12S3
12S3
12S3
12S3
12S3
4C32
9C8
9D4
3C12
6D6
12D6
12D6
12D6
4C3×C6
4C3⋊S3
4C3⋊S3
9SD16
3D12
3C3⋊C8
4C2×C3⋊S3
3GL2(𝔽3)
3Q82S3
3GL2(𝔽3)
3GL2(𝔽3)

Character table of C6.6S4

 class 12A2B3A3B3C3D46A6B6C6D8A8B12
 size 1136288862888181812
ρ1111111111111111    trivial
ρ211-1111111111-1-11    linear of order 2
ρ3220-1-12-12-1-1-1200-1    orthogonal lifted from S3
ρ4220-1-1-122-12-1-100-1    orthogonal lifted from S3
ρ5220-12-1-12-1-12-100-1    orthogonal lifted from S3
ρ62202-1-1-122-1-1-1002    orthogonal lifted from S3
ρ72-202-1-1-10-2111--2-20    complex lifted from GL2(𝔽3)
ρ82-202-1-1-10-2111-2--20    complex lifted from GL2(𝔽3)
ρ93313000-13000-1-1-1    orthogonal lifted from S4
ρ1033-13000-1300011-1    orthogonal lifted from S4
ρ114-40-2-21102-12-1000    orthogonal faithful
ρ124-4041110-4-1-1-1000    orthogonal lifted from GL2(𝔽3)
ρ134-40-211-2022-1-1000    orthogonal faithful
ρ144-40-21-2102-1-12000    orthogonal faithful
ρ15660-3000-2-3000001    orthogonal lifted from C3⋊S4

Permutation representations of C6.6S4
On 24 points - transitive group 24T252
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 16 4 13)(2 17 5 14)(3 18 6 15)(7 19 10 22)(8 20 11 23)(9 21 12 24)
(1 11 4 8)(2 12 5 9)(3 7 6 10)(13 23 16 20)(14 24 17 21)(15 19 18 22)
(7 19 18)(8 20 13)(9 21 14)(10 22 15)(11 23 16)(12 24 17)
(1 4)(2 3)(5 6)(7 9)(10 12)(13 20)(14 19)(15 24)(16 23)(17 22)(18 21)

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

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

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

G:=TransitiveGroup(24,252);

C6.6S4 is a maximal subgroup of
Dic3.5S4  GL2(𝔽3)⋊S3  D6.2S4  S3×GL2(𝔽3)  SL2(𝔽3).D6  C12.14S4  C12.7S4  C322GL2(𝔽3)  C18.6S4  C325GL2(𝔽3)
C6.6S4 is a maximal quotient of
C6.GL2(𝔽3)  C18.6S4  C32.3GL2(𝔽3)  C323GL2(𝔽3)  C325GL2(𝔽3)

Matrix representation of C6.6S4 in GL4(ℚ) generated by

1/21/2-1/21/2
-1/21/21/21/2
1/2-1/21/21/2
-1/2-1/2-1/21/2
,
0010
0001
-1000
0-100
,
000-1
0010
0-100
1000
,
1000
0001
0-100
00-10
,
-1/2-1/21/2-1/2
-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())| [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],[0,0,-1,0,0,0,0,-1,1,0,0,0,0,1,0,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[1,0,0,0,0,0,-1,0,0,0,0,-1,0,1,0,0],[-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] >;

C6.6S4 in GAP, Magma, Sage, TeX

C_6._6S_4
% in TeX

G:=Group("C6.6S4");
// GroupNames label

G:=SmallGroup(144,125);
// by ID

G=gap.SmallGroup(144,125);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-2,49,218,867,1305,117,544,820,202,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=d^3=e^2=1,b^2=c^2=a^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a^3*b,d*b*d^-1=a^3*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^3*c,e*d*e=d^-1>;
// generators/relations

Export

Subgroup lattice of C6.6S4 in TeX
Character table of C6.6S4 in TeX

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