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## G = C3×S3×C3⋊S3order 324 = 22·34

### Direct product of C3, S3 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3×S3×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — C34 — S3×C33 — C3×S3×C3⋊S3
 Lower central C33 — C3×S3×C3⋊S3
 Upper central C1 — C3

Generators and relations for C3×S3×C3⋊S3
G = < a,b,c,d,e,f | a3=b3=c2=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1000 in 224 conjugacy classes, 44 normal (20 characteristic)
C1, C2 [×3], C3 [×2], C3 [×4], C3 [×13], C22, S3, S3 [×13], C6 [×13], C32 [×2], C32 [×8], C32 [×40], D6 [×5], C2×C6, C3×S3, C3×S3 [×4], C3×S3 [×25], C3⋊S3, C3⋊S3 [×9], C3×C6 [×10], C33 [×2], C33 [×4], C33 [×13], S32 [×4], S3×C6 [×5], C2×C3⋊S3, S3×C32, S3×C32 [×4], S3×C32 [×8], C3×C3⋊S3 [×2], C3×C3⋊S3 [×10], C33⋊C2, C32×C6, C34, C3×S32 [×4], S3×C3⋊S3, C6×C3⋊S3, S3×C33, C32×C3⋊S3, C3×C33⋊C2, C3×S3×C3⋊S3
Quotients: C1, C2 [×3], C3, C22, S3 [×5], C6 [×3], D6 [×5], C2×C6, C3×S3 [×5], C3⋊S3, S32 [×4], S3×C6 [×5], C2×C3⋊S3, C3×C3⋊S3, C3×S32 [×4], S3×C3⋊S3, C6×C3⋊S3, C3×S3×C3⋊S3

Smallest permutation representation of C3×S3×C3⋊S3
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 12 14)(2 10 15)(3 11 13)(4 7 36)(5 8 34)(6 9 35)(16 22 19)(17 23 20)(18 24 21)(25 28 31)(26 29 32)(27 30 33)
(1 33)(2 31)(3 32)(4 21)(5 19)(6 20)(7 24)(8 22)(9 23)(10 28)(11 29)(12 30)(13 26)(14 27)(15 25)(16 34)(17 35)(18 36)
(1 13 10)(2 14 11)(3 15 12)(4 34 9)(5 35 7)(6 36 8)(16 23 21)(17 24 19)(18 22 20)(25 30 32)(26 28 33)(27 29 31)
(1 14 12)(2 15 10)(3 13 11)(4 36 7)(5 34 8)(6 35 9)(16 22 19)(17 23 20)(18 24 21)(25 28 31)(26 29 32)(27 30 33)
(1 24)(2 22)(3 23)(4 30)(5 28)(6 29)(7 33)(8 31)(9 32)(10 19)(11 20)(12 21)(13 17)(14 18)(15 16)(25 34)(26 35)(27 36)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3Q 3R ··· 3AC 6A 6B 6C ··· 6N 6O 6P 6Q 6R 6S 6T 6U order 1 2 2 2 3 3 3 ··· 3 3 ··· 3 6 6 6 ··· 6 6 6 6 6 6 6 6 size 1 3 9 27 1 1 2 ··· 2 4 ··· 4 3 3 6 ··· 6 9 9 18 18 18 27 27

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 S3 D6 C3×S3 C3×S3 S3×C6 S32 C3×S32 kernel C3×S3×C3⋊S3 S3×C33 C32×C3⋊S3 C3×C33⋊C2 S3×C3⋊S3 S3×C32 C3×C3⋊S3 C33⋊C2 S3×C32 C3×C3⋊S3 C33 C3×S3 C3⋊S3 C32 C32 C3 # reps 1 1 1 1 2 2 2 2 4 1 5 8 2 10 4 8

Matrix representation of C3×S3×C3⋊S3 in GL6(𝔽7)

 2 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 6 6
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 6 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(7))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,1,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,6,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×S3×C3⋊S3 in GAP, Magma, Sage, TeX

C_3\times S_3\times C_3\rtimes S_3
% in TeX

G:=Group("C3xS3xC3:S3");
// GroupNames label

G:=SmallGroup(324,166);
// by ID

G=gap.SmallGroup(324,166);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,297,1090,7781]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^3=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,c*b*c=b^-1,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

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