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## G = S3×C3≀C3order 486 = 2·35

### Direct product of S3 and C3≀C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×C3≀C3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C3×C3≀C3 — S3×C3≀C3
 Lower central C3 — C32 — C33 — S3×C3≀C3
 Upper central C1 — C3 — C32 — C3≀C3

Generators and relations for S3×C3≀C3
G = < a,b,c,d,e,f | a3=b2=c3=d3=e3=f3=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, cf=fc, de=ed, df=fd, fef-1=cd-1e >

Subgroups: 542 in 123 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3 [×2], C3 [×11], S3, C6 [×6], C9 [×4], C32 [×2], C32 [×30], C18 [×2], C3×S3, C3×S3 [×5], C3×C6 [×6], C3×C9 [×2], He3, He3 [×2], 3- 1+2 [×2], 3- 1+2 [×4], C33 [×2], C33 [×10], S3×C9 [×2], C2×He3, C2×3- 1+2 [×2], S3×C32, S3×C32 [×5], C32×C6, C3≀C3, C3≀C3 [×4], C3×He3, C3×3- 1+2 [×2], C34, C2×C3≀C3, S3×He3, S3×3- 1+2 [×2], S3×C33, C3×C3≀C3, S3×C3≀C3
Quotients: C1, C2, C3 [×4], S3, C6 [×4], C32, C3×S3 [×4], C3×C6, He3, C2×He3, S3×C32, C3≀C3, C2×C3≀C3, S3×He3, S3×C3≀C3

Permutation representations of S3×C3≀C3
On 18 points - transitive group 18T163
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 11)(2 10)(3 12)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)
(1 3 2)(7 8 9)(10 11 12)(16 18 17)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)
(1 4 9)(2 5 7)(3 6 8)(10 13 17)(11 14 18)(12 15 16)
(7 9 8)(16 17 18)

G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11)(2,10)(3,12)(4,14)(5,13)(6,15)(7,17)(8,16)(9,18), (1,3,2)(7,8,9)(10,11,12)(16,18,17), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18), (1,4,9)(2,5,7)(3,6,8)(10,13,17)(11,14,18)(12,15,16), (7,9,8)(16,17,18)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11)(2,10)(3,12)(4,14)(5,13)(6,15)(7,17)(8,16)(9,18), (1,3,2)(7,8,9)(10,11,12)(16,18,17), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18), (1,4,9)(2,5,7)(3,6,8)(10,13,17)(11,14,18)(12,15,16), (7,9,8)(16,17,18) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,11),(2,10),(3,12),(4,14),(5,13),(6,15),(7,17),(8,16),(9,18)], [(1,3,2),(7,8,9),(10,11,12),(16,18,17)], [(1,3,2),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,17,18)], [(1,4,9),(2,5,7),(3,6,8),(10,13,17),(11,14,18),(12,15,16)], [(7,9,8),(16,17,18)])

G:=TransitiveGroup(18,163);

On 27 points - transitive group 27T207
Generators in S27
(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)
(2 3)(5 6)(8 9)(11 12)(14 15)(17 18)(20 21)(23 24)(26 27)
(1 4 7)(2 5 8)(3 6 9)(19 25 22)(20 26 23)(21 27 24)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)
(1 10 22)(2 11 23)(3 12 24)(4 13 25)(5 14 26)(6 15 27)(7 16 19)(8 17 20)(9 18 21)
(19 22 25)(20 23 26)(21 24 27)

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

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), (2,3)(5,6)(8,9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27), (1,4,7)(2,5,8)(3,6,9)(19,25,22)(20,26,23)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,10,22)(2,11,23)(3,12,24)(4,13,25)(5,14,26)(6,15,27)(7,16,19)(8,17,20)(9,18,21), (19,22,25)(20,23,26)(21,24,27) );

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)], [(2,3),(5,6),(8,9),(11,12),(14,15),(17,18),(20,21),(23,24),(26,27)], [(1,4,7),(2,5,8),(3,6,9),(19,25,22),(20,26,23),(21,27,24)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27)], [(1,10,22),(2,11,23),(3,12,24),(4,13,25),(5,14,26),(6,15,27),(7,16,19),(8,17,20),(9,18,21)], [(19,22,25),(20,23,26),(21,24,27)])

G:=TransitiveGroup(27,207);

51 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3M 3N ··· 3U 3V 3W 3X 3Y 6A 6B 6C ··· 6J 6K 6L 9A 9B 9C 9D 9E 9F 9G 9H 18A 18B 18C 18D order 1 2 3 3 3 3 3 3 ··· 3 3 ··· 3 3 3 3 3 6 6 6 ··· 6 6 6 9 9 9 9 9 9 9 9 18 18 18 18 size 1 3 1 1 2 2 2 3 ··· 3 6 ··· 6 9 9 18 18 3 3 9 ··· 9 27 27 9 9 9 9 18 18 18 18 27 27 27 27

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 6 6 type + + + image C1 C2 C3 C3 C3 C6 C6 C6 S3 C3×S3 C3×S3 C3×S3 He3 C2×He3 C3≀C3 C2×C3≀C3 S3×He3 S3×C3≀C3 kernel S3×C3≀C3 C3×C3≀C3 S3×He3 S3×3- 1+2 S3×C33 C3×He3 C3×3- 1+2 C34 C3≀C3 He3 3- 1+2 C33 C3×S3 C32 S3 C3 C3 C1 # reps 1 1 2 4 2 2 4 2 1 2 4 2 2 2 6 6 2 6

Matrix representation of S3×C3≀C3 in GL5(𝔽19)

 18 1 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 18 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 7 11 0 0 0 18 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 9 0 0 0 12 8 7 0 0 11 8 0
,
 7 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 1 7 0 0 0 0 0 1

G:=sub<GL(5,GF(19))| [18,18,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,18,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,7,7,18,0,0,0,11,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,11,12,11,0,0,9,8,8,0,0,0,7,0],[7,0,0,0,0,0,7,0,0,0,0,0,1,1,0,0,0,0,7,0,0,0,0,0,1] >;

S3×C3≀C3 in GAP, Magma, Sage, TeX

S_3\times C_3\wr C_3
% in TeX

G:=Group("S3xC3wrC3");
// GroupNames label

G:=SmallGroup(486,117);
// by ID

G=gap.SmallGroup(486,117);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,303,11669]);
// Polycyclic

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

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