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G = C324S4order 216 = 23·33

2nd semidirect product of C32 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial, rational

Aliases: C324S4, C629S3, C3⋊(C3⋊S4), A4⋊(C3⋊S3), (C3×A4)⋊4S3, (C32×A4)⋊3C2, C22⋊(C33⋊C2), (C2×C6)⋊2(C3⋊S3), SmallGroup(216,165)

Series: Derived Chief Lower central Upper central

C1C22C32×A4 — C324S4
C1C22C2×C6C62C32×A4 — C324S4
C32×A4 — C324S4
C1

Generators and relations for C324S4
 G = < a,b,c,d | a6=b6=c3=d2=1, ab=ba, cac-1=a4b3, dad=a2b3, cbc-1=a3b, dbd=a3b2, dcd=c-1 >

Subgroups: 1060 in 130 conjugacy classes, 35 normal (6 characteristic)
C1, C2 [×2], C3 [×4], C3 [×9], C4, C22, C22, S3 [×13], C6 [×4], D4, C32, C32 [×12], Dic3 [×4], A4 [×9], D6 [×4], C2×C6 [×4], C3⋊S3 [×13], C3×C6, C3⋊D4 [×4], S4 [×9], C33, C3⋊Dic3, C3×A4 [×12], C2×C3⋊S3, C62, C33⋊C2, C327D4, C3⋊S4 [×12], C32×A4, C324S4
Quotients: C1, C2, S3 [×13], C3⋊S3 [×13], S4, C33⋊C2, C3⋊S4 [×4], C324S4

Character table of C324S4

 class 12A2B3A3B3C3D3E3F3G3H3I3J3K3L3M46A6B6C6D
 size 13542222888888888546666
ρ1111111111111111111111    trivial
ρ211-11111111111111-11111    linear of order 2
ρ3220-12-1-1-1-1-1-122-1-120-1-12-1    orthogonal lifted from S3
ρ42202222-1-1-1-1-1-1-1-1-102222    orthogonal lifted from S3
ρ5220-1-12-1-1-12-1-1-1-12202-1-1-1    orthogonal lifted from S3
ρ6220-1-1-12-1-1-1-1-1222-10-12-1-1    orthogonal lifted from S3
ρ72202-1-1-1-1-1-12-1-12-120-1-1-12    orthogonal lifted from S3
ρ8220-1-12-12-1-1-12-12-1-102-1-1-1    orthogonal lifted from S3
ρ9220-1-1-1222-1-1-1-1-1-120-12-1-1    orthogonal lifted from S3
ρ102202-1-1-12-12-1-12-1-1-10-1-1-12    orthogonal lifted from S3
ρ11220-12-1-1-122-1-1-12-1-10-1-12-1    orthogonal lifted from S3
ρ122202-1-1-1-12-1-12-1-12-10-1-1-12    orthogonal lifted from S3
ρ13220-12-1-12-1-12-1-1-12-10-1-12-1    orthogonal lifted from S3
ρ14220-1-12-1-12-12-12-1-1-102-1-1-1    orthogonal lifted from S3
ρ15220-1-1-12-1-1222-1-1-1-10-12-1-1    orthogonal lifted from S3
ρ163-1-133330000000001-1-1-1-1    orthogonal lifted from S4
ρ173-113333000000000-1-1-1-1-1    orthogonal lifted from S4
ρ186-20-3-36-30000000000-2111    orthogonal lifted from C3⋊S4
ρ196-20-3-3-3600000000001-211    orthogonal lifted from C3⋊S4
ρ206-206-3-3-30000000000111-2    orthogonal lifted from C3⋊S4
ρ216-20-36-3-3000000000011-21    orthogonal lifted from C3⋊S4

Smallest permutation representation of C324S4
On 36 points
Generators in S36
(1 2 3 4 5 6)(7 8 9 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 29 12 22 31 16)(2 30 7 23 32 17)(3 25 8 24 33 18)(4 26 9 19 34 13)(5 27 10 20 35 14)(6 28 11 21 36 15)
(2 23 20)(4 19 22)(6 21 24)(7 17 14)(9 13 16)(11 15 18)(25 36 28)(26 29 34)(27 32 30)
(2 24)(3 5)(4 22)(6 20)(7 25)(8 35)(9 29)(10 33)(11 27)(12 31)(13 26)(14 36)(15 30)(16 34)(17 28)(18 32)(21 23)

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

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

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

C324S4 is a maximal subgroup of   C3⋊S3×S4  S3×C3⋊S4
C324S4 is a maximal quotient of   C324CSU2(𝔽3)  C325GL2(𝔽3)  C6210Dic3

Matrix representation of C324S4 in GL7(ℤ)

1000000
0100000
00-1-1000
0010000
00000-11
00000-10
00001-10
,
-1100000
-1000000
0001000
00-1-1000
0000-100
0000-101
0000-110
,
0-100000
1-100000
00-1-1000
0010000
0000001
0000100
0000010
,
-1000000
-1100000
0001000
0010000
0000010
0000100
0000001

G:=sub<GL(7,Integers())| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,-1,0,0,0,0,1,0,0],[-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C324S4 in GAP, Magma, Sage, TeX

C_3^2\rtimes_4S_4
% in TeX

G:=Group("C3^2:4S4");
// GroupNames label

G:=SmallGroup(216,165);
// by ID

G=gap.SmallGroup(216,165);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,49,218,867,3244,1630,1949,2927]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^3=d^2=1,a*b=b*a,c*a*c^-1=a^4*b^3,d*a*d=a^2*b^3,c*b*c^-1=a^3*b,d*b*d=a^3*b^2,d*c*d=c^-1>;
// generators/relations

Export

Character table of C324S4 in TeX

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