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G = C2×S3×S4order 288 = 25·32

Direct product of C2, S3 and S4

direct product, non-abelian, soluble, monomial, rational

Aliases: C2×S3×S4, C23⋊S32, C61(C2×S4), C3⋊S4⋊C22, (C22×C6)⋊D6, (C6×S4)⋊1C2, (C2×A4)⋊1D6, (C3×A4)⋊C23, (S3×A4)⋊C22, (C6×A4)⋊C22, (C3×S4)⋊C22, (C22×S3)⋊D6, C31(C22×S4), A41(C22×S3), (S3×C23)⋊3S3, C22⋊(C2×S32), (C2×S3×A4)⋊5C2, (C2×C3⋊S4)⋊5C2, (C2×C6)⋊(C22×S3), Aut(S3×SL2(𝔽3)), SmallGroup(288,1028)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — C2×S3×S4
C1C22C2×C6C3×A4S3×A4S3×S4 — C2×S3×S4
C3×A4 — C2×S3×S4
C1C2

Generators and relations for C2×S3×S4
 G = < a,b,c,d,e,f,g | a2=b3=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, cbc=b-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 1594 in 272 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2 [×10], C3, C3 [×2], C4 [×4], C22, C22 [×27], S3 [×2], S3 [×10], C6, C6 [×8], C2×C4 [×6], D4 [×16], C23, C23 [×16], C32, Dic3 [×2], C12 [×2], A4, A4, D6, D6 [×27], C2×C6, C2×C6 [×7], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], S4 [×2], S4 [×4], C2×A4, C2×A4 [×3], C22×S3 [×2], C22×S3 [×14], C22×C6, C22×C6, C22×D4, S32 [×4], C3×A4, S3×C6 [×2], C2×C3⋊S3, S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, C2×S4, C2×S4 [×6], C22×A4, S3×C23, S3×C23, C3×S4 [×2], C3⋊S4 [×2], S3×A4 [×2], C2×S32, C6×A4, C2×S3×D4, C22×S4, S3×S4 [×4], C6×S4, C2×C3⋊S4, C2×S3×A4, C2×S3×S4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], S4, C22×S3 [×2], S32, C2×S4 [×3], C2×S32, C22×S4, S3×S4, C2×S3×S4

Character table of C2×S3×S4

 class 12A2B2C2D2E2F2G2H2I2J2K3A3B3C4A4B4C4D6A6B6C6D6E6F6G6H6I12A12B
 size 113333669918182816661818266812121624241212
ρ1111111111111111111111111111111    trivial
ρ21-1-1-1111-11-11-11111-11-1-11-1-11-1-1-11-11    linear of order 2
ρ31-11-11-1-11-111-1111-111-1-11-1-1-11-11-11-1    linear of order 2
ρ411-111-1-1-1-1-111111-1-1111111-1-11-1-1-1-1    linear of order 2
ρ51-1-1-111-111-1-11111-11-11-11-1-1-11-1-111-1    linear of order 2
ρ6111111-1-111-1-1111-1-1-1-11111-1-1111-1-1    linear of order 2
ρ711-111-111-1-1-1-111111-1-11111111-1-111    linear of order 2
ρ81-11-11-11-1-11-111111-1-11-11-1-11-1-11-1-11    linear of order 2
ρ92-2-2-222002-2002-1-10000-22-210011-100    orthogonal lifted from D6
ρ102222220022002-1-10000222-100-1-1-100    orthogonal lifted from S3
ρ112-20-2202-20000-12-12-2001-11-2-111001-1    orthogonal lifted from D6
ρ12220220220000-12-12200-1-1-12-1-1-100-1-1    orthogonal lifted from S3
ρ132-22-22-200-22002-1-10000-22-21001-1100    orthogonal lifted from D6
ρ1422-222-200-2-2002-1-10000222-100-11100    orthogonal lifted from D6
ρ152-20-220-220000-12-1-22001-11-21-1100-11    orthogonal lifted from D6
ρ16220220-2-20000-12-1-2-200-1-1-1211-10011    orthogonal lifted from D6
ρ173-3-31-13-11-11-113001-11-1-3-110-11000-11    orthogonal lifted from C2×S4
ρ1833-3-1-1-31111-1-1300-1-1113-1-1011000-1-1    orthogonal lifted from C2×S4
ρ19333-1-13-1-1-1-1-1-130011113-1-10-1-100011    orthogonal lifted from S4
ρ203-331-1-31-11-1-11300-111-1-3-1101-10001-1    orthogonal lifted from C2×S4
ρ213-3-31-131-1-111-1300-11-11-3-1101-10001-1    orthogonal lifted from C2×S4
ρ2233-3-1-1-3-1-1111130011-1-13-1-10-1-100011    orthogonal lifted from C2×S4
ρ23333-1-1311-1-111300-1-1-1-13-1-1011000-1-1    orthogonal lifted from S4
ρ243-331-1-3-111-11-13001-1-11-3-110-11000-11    orthogonal lifted from C2×S4
ρ254-40-440000000-2-2100002-22200-10000    orthogonal lifted from C2×S32
ρ26440440000000-2-210000-2-2-2-20010000    orthogonal lifted from S32
ρ27660-2-20220000-300-2-200-3110-1-100011    orthogonal lifted from S3×S4
ρ286-602-202-20000-300-220031-10-11000-11    orthogonal faithful
ρ296-602-20-220000-3002-20031-101-10001-1    orthogonal faithful
ρ30660-2-20-2-20000-3002200-311011000-1-1    orthogonal lifted from S3×S4

Permutation representations of C2×S3×S4
On 18 points - transitive group 18T111
Generators in S18
(1 9)(2 7)(3 8)(4 18)(5 16)(6 17)(10 14)(11 15)(12 13)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(2 3)(4 5)(7 8)(10 12)(13 14)(16 18)
(1 9)(2 7)(3 8)(10 14)(11 15)(12 13)
(4 18)(5 16)(6 17)(10 14)(11 15)(12 13)
(1 11 17)(2 12 18)(3 10 16)(4 7 13)(5 8 14)(6 9 15)
(4 13)(5 14)(6 15)(10 16)(11 17)(12 18)

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

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

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

G:=TransitiveGroup(18,111);

On 24 points - transitive group 24T679
Generators in S24
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)
(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 7)(2 9)(3 8)(4 20)(5 19)(6 21)(10 24)(11 23)(12 22)(13 17)(14 16)(15 18)
(1 14)(2 15)(3 13)(4 24)(5 22)(6 23)(7 16)(8 17)(9 18)(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)
(4 24 17)(5 22 18)(6 23 16)(10 13 20)(11 14 21)(12 15 19)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 18)(11 16)(12 17)(13 22)(14 23)(15 24)

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

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

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

G:=TransitiveGroup(24,679);

Matrix representation of C2×S3×S4 in GL5(ℤ)

-10000
0-1000
00-100
000-10
0000-1
,
-1-1000
10000
00100
00010
00001
,
10000
-1-1000
00-100
000-10
0000-1
,
10000
01000
00-100
000-10
00001
,
10000
01000
00100
000-10
0000-1
,
10000
01000
00001
00100
00010
,
10000
01000
00-100
0000-1
000-10

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,1,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,-1,0,0,0,0,-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,-1,0,0,0,0,0,-1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,-1,0] >;

C2×S3×S4 in GAP, Magma, Sage, TeX

C_2\times S_3\times S_4
% in TeX

G:=Group("C2xS3xS4");
// GroupNames label

G:=SmallGroup(288,1028);
// by ID

G=gap.SmallGroup(288,1028);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,234,1684,3036,782,1777,1350]);
// Polycyclic

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

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Character table of C2×S3×S4 in TeX

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