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G = S33order 216 = 23·33

Direct product of S3, S3 and S3

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: S33, C33⋊C23, C3⋊S32D6, (C3×S3)⋊1D6, (S3×C32)⋊C22, C33⋊C2⋊C22, C324D63C2, C324(C22×S3), (S3×C3⋊S3)⋊C2, C31(C2×S32), (C3×S32)⋊3C2, (C3×C3⋊S3)⋊C22, Hol(C3×S3), SmallGroup(216,162)

Series: Derived Chief Lower central Upper central

C1C33 — S33
C1C3C32C33S3×C32C3×S32 — S33
C33 — S33
C1

Generators and relations for S33
 G = < a,b,c,d,e,f | a3=b2=c3=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 904 in 162 conjugacy classes, 38 normal (5 characteristic)
C1, C2 [×7], C3 [×3], C3 [×4], C22 [×7], S3 [×3], S3 [×16], C6 [×12], C23, C32 [×3], C32 [×4], D6 [×21], C2×C6 [×3], C3×S3 [×6], C3×S3 [×12], C3⋊S3 [×3], C3⋊S3 [×7], C3×C6 [×3], C22×S3 [×3], C33, S32 [×3], S32 [×12], S3×C6 [×6], C2×C3⋊S3 [×3], S3×C32 [×3], C3×C3⋊S3 [×3], C33⋊C2, C2×S32 [×3], C3×S32 [×3], S3×C3⋊S3 [×3], C324D6, S33
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C22×S3 [×3], S32 [×3], C2×S32 [×3], S33

Character table of S33

 class 12A2B2C2D2E2F2G3A3B3C3D3E3F3G6A6B6C6D6E6F6G6H6I6J6K6L
 size 1333999272224448666666121212181818
ρ1111111111111111111111111111    trivial
ρ21-1-1-1111-11111111-1-1-1-1-1-1-1-1-1111    linear of order 2
ρ31-11-1-11-1111111111-1-1-1-11-11-1-1-11    linear of order 2
ρ411-11-11-1-11111111-11111-11-11-1-11    linear of order 2
ρ51-1111-1-1-111111111-111-1111-1-11-1    linear of order 2
ρ611-1-11-1-111111111-11-1-11-1-1-11-11-1    linear of order 2
ρ7111-1-1-11-1111111111-1-111-1111-1-1    linear of order 2
ρ81-1-11-1-1111111111-1-111-1-11-1-11-1-1    linear of order 2
ρ92-22000-20-122-1-12-12-2001-10-11100    orthogonal lifted from D6
ρ1020-2-2200022-12-1-1-110-210-21100-10    orthogonal lifted from D6
ρ11220202002-12-12-1-10-1-1220-10-100-1    orthogonal lifted from S3
ρ122022200022-12-1-1-1-102-102-1-100-10    orthogonal lifted from S3
ρ132-20-202002-12-12-1-1011-2-2010100-1    orthogonal lifted from D6
ρ142-2-200020-122-1-12-1-2-20011011-100    orthogonal lifted from D6
ρ1520-22-200022-12-1-1-1102-10-2-110010    orthogonal lifted from D6
ρ1622-2000-20-122-1-12-1-2200-1101-1100    orthogonal lifted from D6
ρ172-2020-2002-12-12-1-101-12-20-101001    orthogonal lifted from D6
ρ1822200020-122-1-12-12200-1-10-1-1-100    orthogonal lifted from S3
ρ19220-20-2002-12-12-1-10-11-22010-1001    orthogonal lifted from D6
ρ20202-2-200022-12-1-1-1-10-21021-10010    orthogonal lifted from D6
ρ2140400000-24-2-21-21-20000-2010000    orthogonal lifted from S32
ρ22400-400004-2-2-2-211002200-100000    orthogonal lifted from C2×S32
ρ2340-400000-24-2-21-212000020-10000    orthogonal lifted from C2×S32
ρ24400400004-2-2-2-21100-2-200100000    orthogonal lifted from S32
ρ254-4000000-2-241-2-2102002000-1000    orthogonal lifted from C2×S32
ρ2644000000-2-241-2-210-200-20001000    orthogonal lifted from S32
ρ2780000000-4-4-4222-1000000000000    orthogonal faithful

Permutation representations of S33
On 12 points - transitive group 12T117
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 4)(2 6)(3 5)(7 11)(8 10)(9 12)
(1 3 2)(4 5 6)(7 9 8)(10 11 12)
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)

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

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,4)(2,6)(3,5)(7,11)(8,10)(9,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12), (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,2,3)(4,6,5)(7,9,8)(10,11,12), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,4),(2,6),(3,5),(7,11),(8,10),(9,12)], [(1,3,2),(4,5,6),(7,9,8),(10,11,12)], [(1,10),(2,11),(3,12),(4,8),(5,9),(6,7)], [(1,2,3),(4,6,5),(7,9,8),(10,11,12)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12)])

G:=TransitiveGroup(12,117);

On 18 points - transitive group 18T96
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 10)(2 12)(3 11)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)
(1 10)(2 11)(3 12)(4 15)(5 13)(6 14)(7 18)(8 16)(9 17)
(1 5 8)(2 6 9)(3 4 7)(10 13 16)(11 14 17)(12 15 18)
(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)

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

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

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

G:=TransitiveGroup(18,96);

On 24 points - transitive group 24T557
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 4)(2 6)(3 5)(7 11)(8 10)(9 12)(13 17)(14 16)(15 18)(19 23)(20 22)(21 24)
(1 3 2)(4 5 6)(7 9 8)(10 11 12)(13 14 15)(16 18 17)(19 20 21)(22 24 23)
(1 14)(2 15)(3 13)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 18 17)(19 21 20)(22 23 24)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

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

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

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

G:=TransitiveGroup(24,557);

On 27 points - transitive group 27T86
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)(10 12)(14 15)(17 18)(19 21)(23 24)(26 27)
(1 13 25)(2 14 26)(3 15 27)(4 20 7)(5 21 8)(6 19 9)(10 18 24)(11 16 22)(12 17 23)
(4 20)(5 21)(6 19)(10 24)(11 22)(12 23)(13 25)(14 26)(15 27)
(1 7 16)(2 8 17)(3 9 18)(4 22 13)(5 23 14)(6 24 15)(10 27 19)(11 25 20)(12 26 21)
(4 22)(5 23)(6 24)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)

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

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

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

G:=TransitiveGroup(27,86);

S33 is a maximal quotient of
Dic36S32  D64S32  D6⋊S32  (S3×C6)⋊D6  C3⋊S34D12  C335(C2×Q8)  C336(C2×Q8)  (S3×C6).D6  D6.S32  D6.4S32  D6.3S32  D6⋊S3⋊S3  D6.6S32  Dic3.S32

Polynomial with Galois group S33 over ℚ
actionf(x)Disc(f)
12T117x12-4x9+2x6+4x3-2226·318

Matrix representation of S33 in GL6(ℤ)

100000
010000
00-1100
00-1000
000010
000001
,
100000
010000
000100
001000
0000-10
00000-1
,
100000
010000
001000
000100
0000-11
0000-10
,
-100000
0-10000
00-1000
000-100
00000-1
0000-10
,
-110000
-100000
001000
000100
000010
000001
,
0-10000
-100000
00-1000
000-100
000010
000001

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[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,-1,-1,0,0,0,0,1,0],[-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,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S33 in GAP, Magma, Sage, TeX

S_3^3
% in TeX

G:=Group("S3^3");
// GroupNames label

G:=SmallGroup(216,162);
// by ID

G=gap.SmallGroup(216,162);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,111,730,5189]);
// Polycyclic

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

Export

Character table of S33 in TeX

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