Copied to
clipboard

G = C3×C3⋊S4order 216 = 23·33

Direct product of C3 and C3⋊S4

direct product, non-abelian, soluble, monomial

Aliases: C3×C3⋊S4, C628S3, C323S4, C3⋊(C3×S4), A4⋊(C3×S3), (C3×A4)⋊3S3, (C3×A4)⋊4C6, (C32×A4)⋊2C2, C22⋊(C3×C3⋊S3), (C2×C6)⋊2(C3×S3), (C2×C6)⋊1(C3⋊S3), SmallGroup(216,164)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — C3×C3⋊S4
C1C22C2×C6C3×A4C32×A4 — C3×C3⋊S4
C3×A4 — C3×C3⋊S4
C1C3

Generators and relations for C3×C3⋊S4
 G = < a,b,c,d,e,f | a3=b3=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 360 in 77 conjugacy classes, 18 normal (12 characteristic)
C1, C2 [×2], C3 [×2], C3 [×7], C4, C22, C22, S3 [×4], C6 [×4], D4, C32, C32 [×8], Dic3, C12, A4 [×3], A4 [×3], D6, C2×C6 [×2], C2×C6 [×2], C3×S3 [×4], C3⋊S3, C3×C6, C3⋊D4, C3×D4, S4 [×3], C33, C3×Dic3, C3×A4, C3×A4 [×3], C3×A4 [×4], S3×C6, C62, C3×C3⋊S3, C3×C3⋊D4, C3×S4 [×3], C3⋊S4, C32×A4, C3×C3⋊S4
Quotients: C1, C2, C3, S3 [×4], C6, C3×S3 [×4], C3⋊S3, S4, C3×C3⋊S3, C3×S4, C3⋊S4, C3×C3⋊S4

Character table of C3×C3⋊S4

 class 12A2B3A3B3C3D3E3F3G3H3I3J3K3L3M3N46A6B6C6D6E6F6G12A12B
 size 131811222888888888183366618181818
ρ1111111111111111111111111111    trivial
ρ211-111111111111111-111111-1-1-1-1    linear of order 2
ρ3111ζ3ζ32ζ32ζ31ζ3ζ3111ζ32ζ32ζ32ζ31ζ3ζ32ζ321ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ411-1ζ3ζ32ζ32ζ31ζ3ζ3111ζ32ζ32ζ32ζ3-1ζ3ζ32ζ321ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ511-1ζ32ζ3ζ3ζ321ζ32ζ32111ζ3ζ3ζ3ζ32-1ζ32ζ3ζ31ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ6111ζ32ζ3ζ3ζ321ζ32ζ32111ζ3ζ3ζ3ζ321ζ32ζ3ζ31ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ722022-1-1-1-1-1-12-12-1-12022-1-1-10000    orthogonal lifted from S3
ρ822022-1-1-1-12-1-12-12-1-1022-1-1-10000    orthogonal lifted from S3
ρ922022222-1-1-1-1-1-1-1-1-10222220000    orthogonal lifted from S3
ρ1022022-1-1-12-12-1-1-1-12-1022-1-1-10000    orthogonal lifted from S3
ρ11220-1+-3-1--3ζ6ζ65-1ζ65-1+-3-1-12ζ6-1--3ζ6ζ650-1+-3-1--3ζ6-1ζ650000    complex lifted from C3×S3
ρ12220-1+-3-1--3-1--3-1+-32ζ65ζ65-1-1-1ζ6ζ6ζ6ζ650-1+-3-1--3-1--32-1+-30000    complex lifted from C3×S3
ρ13220-1--3-1+-3-1+-3-1--32ζ6ζ6-1-1-1ζ65ζ65ζ65ζ60-1--3-1+-3-1+-32-1--30000    complex lifted from C3×S3
ρ14220-1+-3-1--3ζ6ζ65-1-1+-3ζ652-1-1ζ6ζ6-1--3ζ650-1+-3-1--3ζ6-1ζ650000    complex lifted from C3×S3
ρ15220-1--3-1+-3ζ65ζ6-1ζ6ζ6-12-1-1+-3ζ65ζ65-1--30-1--3-1+-3ζ65-1ζ60000    complex lifted from C3×S3
ρ16220-1--3-1+-3ζ65ζ6-1-1--3ζ62-1-1ζ65ζ65-1+-3ζ60-1--3-1+-3ζ65-1ζ60000    complex lifted from C3×S3
ρ17220-1--3-1+-3ζ65ζ6-1ζ6-1--3-1-12ζ65-1+-3ζ65ζ60-1--3-1+-3ζ65-1ζ60000    complex lifted from C3×S3
ρ18220-1+-3-1--3ζ6ζ65-1ζ65ζ65-12-1-1--3ζ6ζ6-1+-30-1+-3-1--3ζ6-1ζ650000    complex lifted from C3×S3
ρ193-1-1333330000000001-1-1-1-1-1-1-111    orthogonal lifted from S4
ρ203-1133333000000000-1-1-1-1-1-111-1-1    orthogonal lifted from S4
ρ213-1-1-3+3-3/2-3-3-3/2-3-3-3/2-3+3-3/230000000001ζ65ζ6ζ6-1ζ65ζ65ζ6ζ3ζ32    complex lifted from C3×S4
ρ223-11-3+3-3/2-3-3-3/2-3-3-3/2-3+3-3/23000000000-1ζ65ζ6ζ6-1ζ65ζ3ζ32ζ65ζ6    complex lifted from C3×S4
ρ233-11-3-3-3/2-3+3-3/2-3+3-3/2-3-3-3/23000000000-1ζ6ζ65ζ65-1ζ6ζ32ζ3ζ6ζ65    complex lifted from C3×S4
ρ243-1-1-3-3-3/2-3+3-3/2-3+3-3/2-3-3-3/230000000001ζ6ζ65ζ65-1ζ6ζ6ζ65ζ32ζ3    complex lifted from C3×S4
ρ256-2066-3-3-30000000000-2-21110000    orthogonal lifted from C3⋊S4
ρ266-20-3+3-3-3-3-33+3-3/23-3-3/2-300000000001--31+-3ζ321ζ30000    complex faithful
ρ276-20-3-3-3-3+3-33-3-3/23+3-3/2-300000000001+-31--3ζ31ζ320000    complex faithful

Permutation representations of C3×C3⋊S4
On 24 points - transitive group 24T564
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 3 2)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 17 18)(19 21 20)(22 23 24)
(1 14)(2 15)(3 13)(4 17)(5 18)(6 16)(7 23)(8 24)(9 22)(10 20)(11 21)(12 19)
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(1 2 3)(4 16 22)(5 17 23)(6 18 24)(7 9 8)(10 14 19)(11 15 20)(12 13 21)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)

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

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

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

G:=TransitiveGroup(24,564);

C3×C3⋊S4 is a maximal subgroup of   C3×S3×S4  C6210D6

Matrix representation of C3×C3⋊S4 in GL5(𝔽13)

30000
03000
00900
00090
00009
,
012000
112000
00100
00010
00001
,
10000
01000
000121
000120
001120
,
10000
01000
001200
001201
001210
,
121000
120000
001012
000012
000112
,
012000
120000
00100
00001
00010

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[0,1,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[12,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,12,12],[0,12,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×C3⋊S4 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes S_4
% in TeX

G:=Group("C3xC3:S4");
// GroupNames label

G:=SmallGroup(216,164);
// by ID

G=gap.SmallGroup(216,164);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,218,867,3244,556,1949,989]);
// Polycyclic

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

Export

Character table of C3×C3⋊S4 in TeX

׿
×
𝔽