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G = A4xC3:S3order 216 = 23·33

Direct product of A4 and C3:S3

direct product, metabelian, soluble, monomial, A-group

Aliases: A4xC3:S3, C62:6C6, C3:(S3xA4), (C3xA4):5S3, C32:4(C2xA4), (C32xA4):5C2, (C2xC6):3(C3xS3), C22:2(C3xC3:S3), (C22xC3:S3):3C3, SmallGroup(216,167)

Series: Derived Chief Lower central Upper central

C1C62 — A4xC3:S3
C1C3C32C62C32xA4 — A4xC3:S3
C62 — A4xC3:S3
C1

Generators and relations for A4xC3:S3
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f2=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 492 in 88 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C3, C3, C22, C22, S3, C6, C23, C32, C32, A4, A4, D6, C2xC6, C3xS3, C3:S3, C3:S3, C3xC6, C2xA4, C22xS3, C33, C3xA4, C3xA4, C2xC3:S3, C62, C3xC3:S3, S3xA4, C22xC3:S3, C32xA4, A4xC3:S3
Quotients: C1, C2, C3, S3, C6, A4, C3xS3, C3:S3, C2xA4, C3xC3:S3, S3xA4, A4xC3:S3

Character table of A4xC3:S3

 class 12A2B2C3A3B3C3D3E3F3G3H3I3J3K3L3M3N6A6B6C6D6E6F
 size 139272222448888888866663636
ρ1111111111111111111111111    trivial
ρ211-1-1111111111111111111-1-1    linear of order 2
ρ311111111ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ321111ζ32ζ3    linear of order 3
ρ411-1-11111ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ31111ζ65ζ6    linear of order 6
ρ511-1-11111ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ321111ζ6ζ65    linear of order 6
ρ611111111ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ31111ζ3ζ32    linear of order 3
ρ72200-1-1-1222-1-1-1-1-12-12-1-1-1200    orthogonal lifted from S3
ρ82200-12-1-122-122-1-1-1-1-12-1-1-100    orthogonal lifted from S3
ρ922002-1-1-122-1-1-1-12-12-1-1-12-100    orthogonal lifted from S3
ρ102200-1-12-1222-1-12-1-1-1-1-12-1-100    orthogonal lifted from S3
ρ112200-1-1-12-1--3-1+-3ζ65ζ65ζ6ζ6ζ6-1--3ζ65-1+-3-1-1-1200    complex lifted from C3xS3
ρ122200-12-1-1-1+-3-1--3ζ6-1--3-1+-3ζ65ζ65ζ65ζ6ζ62-1-1-100    complex lifted from C3xS3
ρ132200-1-12-1-1--3-1+-3-1+-3ζ65ζ6-1--3ζ6ζ6ζ65ζ65-12-1-100    complex lifted from C3xS3
ρ1422002-1-1-1-1--3-1+-3ζ65ζ65ζ6ζ6-1--3ζ6-1+-3ζ65-1-12-100    complex lifted from C3xS3
ρ152200-12-1-1-1--3-1+-3ζ65-1+-3-1--3ζ6ζ6ζ6ζ65ζ652-1-1-100    complex lifted from C3xS3
ρ162200-1-1-12-1+-3-1--3ζ6ζ6ζ65ζ65ζ65-1+-3ζ6-1--3-1-1-1200    complex lifted from C3xS3
ρ172200-1-12-1-1+-3-1--3-1--3ζ6ζ65-1+-3ζ65ζ65ζ6ζ6-12-1-100    complex lifted from C3xS3
ρ1822002-1-1-1-1+-3-1--3ζ6ζ6ζ65ζ65-1+-3ζ65-1--3ζ6-1-12-100    complex lifted from C3xS3
ρ193-13-133330000000000-1-1-1-100    orthogonal lifted from A4
ρ203-1-3133330000000000-1-1-1-100    orthogonal lifted from C2xA4
ρ216-2006-3-3-3000000000011-2100    orthogonal lifted from S3xA4
ρ226-200-3-3-360000000000111-200    orthogonal lifted from S3xA4
ρ236-200-36-3-30000000000-211100    orthogonal lifted from S3xA4
ρ246-200-3-36-300000000001-21100    orthogonal lifted from S3xA4

Smallest permutation representation of A4xC3:S3
On 36 points
Generators in S36
(1 18)(2 22)(3 11)(4 29)(5 31)(6 25)(7 20)(8 13)(9 34)(10 12)(14 15)(16 17)(19 21)(23 24)(26 27)(28 30)(32 33)(35 36)
(1 16)(2 23)(3 12)(4 30)(5 32)(6 26)(7 21)(8 14)(9 35)(10 11)(13 15)(17 18)(19 20)(22 24)(25 27)(28 29)(31 33)(34 36)
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 3 2)(4 6 5)(7 9 8)(10 24 17)(11 22 18)(12 23 16)(13 20 34)(14 21 35)(15 19 36)(25 31 29)(26 32 30)(27 33 28)
(1 4 7)(2 5 8)(3 6 9)(10 27 36)(11 25 34)(12 26 35)(13 22 31)(14 23 32)(15 24 33)(16 30 21)(17 28 19)(18 29 20)
(2 3)(4 7)(5 9)(6 8)(10 24)(11 22)(12 23)(13 25)(14 26)(15 27)(19 28)(20 29)(21 30)(31 34)(32 35)(33 36)

G:=sub<Sym(36)| (1,18)(2,22)(3,11)(4,29)(5,31)(6,25)(7,20)(8,13)(9,34)(10,12)(14,15)(16,17)(19,21)(23,24)(26,27)(28,30)(32,33)(35,36), (1,16)(2,23)(3,12)(4,30)(5,32)(6,26)(7,21)(8,14)(9,35)(10,11)(13,15)(17,18)(19,20)(22,24)(25,27)(28,29)(31,33)(34,36), (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,3,2)(4,6,5)(7,9,8)(10,24,17)(11,22,18)(12,23,16)(13,20,34)(14,21,35)(15,19,36)(25,31,29)(26,32,30)(27,33,28), (1,4,7)(2,5,8)(3,6,9)(10,27,36)(11,25,34)(12,26,35)(13,22,31)(14,23,32)(15,24,33)(16,30,21)(17,28,19)(18,29,20), (2,3)(4,7)(5,9)(6,8)(10,24)(11,22)(12,23)(13,25)(14,26)(15,27)(19,28)(20,29)(21,30)(31,34)(32,35)(33,36)>;

G:=Group( (1,18)(2,22)(3,11)(4,29)(5,31)(6,25)(7,20)(8,13)(9,34)(10,12)(14,15)(16,17)(19,21)(23,24)(26,27)(28,30)(32,33)(35,36), (1,16)(2,23)(3,12)(4,30)(5,32)(6,26)(7,21)(8,14)(9,35)(10,11)(13,15)(17,18)(19,20)(22,24)(25,27)(28,29)(31,33)(34,36), (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,3,2)(4,6,5)(7,9,8)(10,24,17)(11,22,18)(12,23,16)(13,20,34)(14,21,35)(15,19,36)(25,31,29)(26,32,30)(27,33,28), (1,4,7)(2,5,8)(3,6,9)(10,27,36)(11,25,34)(12,26,35)(13,22,31)(14,23,32)(15,24,33)(16,30,21)(17,28,19)(18,29,20), (2,3)(4,7)(5,9)(6,8)(10,24)(11,22)(12,23)(13,25)(14,26)(15,27)(19,28)(20,29)(21,30)(31,34)(32,35)(33,36) );

G=PermutationGroup([[(1,18),(2,22),(3,11),(4,29),(5,31),(6,25),(7,20),(8,13),(9,34),(10,12),(14,15),(16,17),(19,21),(23,24),(26,27),(28,30),(32,33),(35,36)], [(1,16),(2,23),(3,12),(4,30),(5,32),(6,26),(7,21),(8,14),(9,35),(10,11),(13,15),(17,18),(19,20),(22,24),(25,27),(28,29),(31,33),(34,36)], [(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,3,2),(4,6,5),(7,9,8),(10,24,17),(11,22,18),(12,23,16),(13,20,34),(14,21,35),(15,19,36),(25,31,29),(26,32,30),(27,33,28)], [(1,4,7),(2,5,8),(3,6,9),(10,27,36),(11,25,34),(12,26,35),(13,22,31),(14,23,32),(15,24,33),(16,30,21),(17,28,19),(18,29,20)], [(2,3),(4,7),(5,9),(6,8),(10,24),(11,22),(12,23),(13,25),(14,26),(15,27),(19,28),(20,29),(21,30),(31,34),(32,35),(33,36)]])

A4xC3:S3 is a maximal subgroup of   C62:Dic3  C62:10D6  S32xA4
A4xC3:S3 is a maximal quotient of   C3:Dic3.2A4

Matrix representation of A4xC3:S3 in GL7(F7)

1000000
0100000
0010000
0001000
0000600
0000601
0000610
,
1000000
0100000
0010000
0001000
0000061
0000060
0000160
,
4000000
0400000
0040000
0004000
0000001
0000100
0000010
,
6100000
6000000
0010000
0001000
0000100
0000010
0000001
,
0600000
1600000
0066000
0010000
0000100
0000010
0000001
,
6100000
0100000
0001000
0010000
0000100
0000010
0000001

G:=sub<GL(7,GF(7))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,6,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,6,6,6,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[6,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

A4xC3:S3 in GAP, Magma, Sage, TeX

A_4\times C_3\rtimes S_3
% in TeX

G:=Group("A4xC3:S3");
// GroupNames label

G:=SmallGroup(216,167);
// by ID

G=gap.SmallGroup(216,167);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-3,170,81,1444,5189]);
// Polycyclic

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

Export

Character table of A4xC3:S3 in TeX

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