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G = S3×C3.A4order 216 = 23·33

Direct product of S3 and C3.A4

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

Aliases: S3×C3.A4, C62.3C6, (C2×C6)⋊C18, (C3×S3).A4, (C22×S3)⋊C9, C3.4(S3×A4), C222(S3×C9), C32.2(C2×A4), C3⋊(C2×C3.A4), (S3×C2×C6).C3, (C3×C3.A4)⋊1C2, (C2×C6).8(C3×S3), SmallGroup(216,98)

Series: Derived Chief Lower central Upper central

C1C2×C6 — S3×C3.A4
C1C3C2×C6C62C3×C3.A4 — S3×C3.A4
C2×C6 — S3×C3.A4
C1C3

Generators and relations for S3×C3.A4
 G = < a,b,c,d,e,f | a3=b2=c3=d2=e2=1, f3=c, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

3C2
3C2
9C2
2C3
9C22
9C22
3C6
3C6
3C6
3S3
6C6
9C6
4C9
8C9
3C23
2C2×C6
3D6
3D6
9C2×C6
9C2×C6
3C3×C6
3C3×S3
12C18
4C3×C9
3C22×C6
2C3.A4
3S3×C6
3S3×C6
4S3×C9
3C2×C3.A4

Smallest permutation representation of S3×C3.A4
On 36 points
Generators in S36
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 28)(8 29)(9 30)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)
(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)(28 31 34)(29 32 35)(30 33 36)
(1 20)(2 21)(4 23)(5 24)(7 26)(8 27)(10 31)(11 32)(13 34)(14 35)(16 28)(17 29)
(2 21)(3 22)(5 24)(6 25)(8 27)(9 19)(11 32)(12 33)(14 35)(15 36)(17 29)(18 30)
(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)

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

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

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

S3×C3.A4 is a maximal quotient of   Q8⋊C93S3

36 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I9A···9F9G···9L18A···18F
order1222333336666666669···99···918···18
size1339112223333666994···48···812···12

36 irreducible representations

dim111111222333366
type++++++
imageC1C2C3C6C9C18S3C3×S3S3×C9A4C2×A4C3.A4C2×C3.A4S3×A4S3×C3.A4
kernelS3×C3.A4C3×C3.A4S3×C2×C6C62C22×S3C2×C6C3.A4C2×C6C22C3×S3C32S3C3C3C1
# reps112266126112212

Matrix representation of S3×C3.A4 in GL5(𝔽19)

77000
011000
00100
00010
00001
,
180000
131000
00100
00010
00001
,
10000
01000
00700
00070
00007
,
10000
01000
001800
001810
000018
,
10000
01000
00100
001180
001018
,
10000
01000
004110
000154
000150

G:=sub<GL(5,GF(19))| [7,0,0,0,0,7,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,13,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,18,18,0,0,0,0,1,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,11,15,15,0,0,0,4,0] >;

S3×C3.A4 in GAP, Magma, Sage, TeX

S_3\times C_3.A_4
% in TeX

G:=Group("S3xC3.A4");
// GroupNames label

G:=SmallGroup(216,98);
// by ID

G=gap.SmallGroup(216,98);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-3,43,657,280,5189]);
// Polycyclic

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

Export

Subgroup lattice of S3×C3.A4 in TeX

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