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## G = C3×C22⋊S4order 288 = 25·32

### Direct product of C3 and C22⋊S4

Aliases: C3×C22⋊S4, (C2×C6)⋊1S4, C22⋊(C3×S4), C22⋊A44C6, (C23×C6)⋊3S3, C245(C3×S3), (C3×C22⋊A4)⋊2C2, SmallGroup(288,1035)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C22⋊A4 — C3×C22⋊S4
 Chief series C1 — C22 — C24 — C22⋊A4 — C3×C22⋊A4 — C3×C22⋊S4
 Lower central C22⋊A4 — C3×C22⋊S4
 Upper central C1 — C3

Generators and relations for C3×C22⋊S4
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fcf-1=bc=cb, bd=db, be=eb, fbf-1=gbg=c, cd=dc, ce=ec, gcg=b, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 574 in 119 conjugacy classes, 14 normal (8 characteristic)
C1, C2 [×5], C3, C3 [×2], C4 [×3], C22 [×3], C22 [×11], S3, C6 [×5], C2×C4 [×3], D4 [×6], C23 [×5], C32, C12 [×3], A4 [×9], C2×C6 [×3], C2×C6 [×11], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3, C2×C12 [×3], C3×D4 [×6], S4 [×3], C22×C6 [×5], C22≀C2, C3×A4 [×4], C3×C22⋊C4 [×3], C6×D4 [×3], C22⋊A4, C22⋊A4, C23×C6, C3×S4 [×3], C3×C22≀C2, C22⋊S4, C3×C22⋊A4, C3×C22⋊S4
Quotients: C1, C2, C3, S3, C6, C3×S3, S4 [×3], C3×S4 [×3], C22⋊S4, C3×C22⋊S4

Character table of C3×C22⋊S4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C 12D 12E 12F size 1 3 3 3 6 12 1 1 32 32 32 12 12 12 3 3 3 3 3 3 6 6 12 12 12 12 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 ζ32 ζ3 ζ32 ζ3 1 -1 -1 -1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ65 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ4 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ5 1 1 1 1 1 -1 ζ3 ζ32 ζ3 ζ32 1 -1 -1 -1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ6 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ6 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ7 2 2 2 2 2 0 2 2 -1 -1 -1 0 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 2 2 2 0 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 0 -1+√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ9 2 2 2 2 2 0 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 0 -1-√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ10 3 -1 -1 3 -1 -1 3 3 0 0 0 1 -1 1 -1 -1 3 -1 3 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from S4 ρ11 3 -1 3 -1 -1 1 3 3 0 0 0 -1 -1 1 -1 -1 -1 3 -1 3 -1 -1 1 1 1 -1 -1 -1 1 -1 orthogonal lifted from S4 ρ12 3 3 -1 -1 -1 -1 3 3 0 0 0 -1 1 1 3 3 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 orthogonal lifted from S4 ρ13 3 -1 -1 3 -1 1 3 3 0 0 0 -1 1 -1 -1 -1 3 -1 3 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 orthogonal lifted from S4 ρ14 3 -1 3 -1 -1 -1 3 3 0 0 0 1 1 -1 -1 -1 -1 3 -1 3 -1 -1 -1 -1 -1 1 1 1 -1 1 orthogonal lifted from S4 ρ15 3 3 -1 -1 -1 1 3 3 0 0 0 1 -1 -1 3 3 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 -1 1 orthogonal lifted from S4 ρ16 3 -1 -1 3 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 1 -1 ζ6 ζ65 -3-3√-3/2 ζ6 -3+3√-3/2 ζ65 ζ65 ζ6 ζ32 ζ3 ζ6 ζ6 ζ32 ζ3 ζ65 ζ65 complex lifted from C3×S4 ρ17 3 -1 3 -1 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 -1 1 ζ65 ζ6 ζ65 -3+3√-3/2 ζ6 -3-3√-3/2 ζ6 ζ65 ζ3 ζ32 ζ3 ζ65 ζ65 ζ6 ζ32 ζ6 complex lifted from C3×S4 ρ18 3 3 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 1 1 -3-3√-3/2 -3+3√-3/2 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 ζ65 ζ32 ζ6 ζ32 ζ3 ζ3 ζ65 complex lifted from C3×S4 ρ19 3 3 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 1 1 -3+3√-3/2 -3-3√-3/2 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 ζ6 ζ3 ζ65 ζ3 ζ32 ζ32 ζ6 complex lifted from C3×S4 ρ20 3 3 -1 -1 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 1 -1 -1 -3+3√-3/2 -3-3√-3/2 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 ζ3 ζ32 ζ65 ζ3 ζ65 ζ6 ζ6 ζ32 complex lifted from C3×S4 ρ21 3 3 -1 -1 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 1 -1 -1 -3-3√-3/2 -3+3√-3/2 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 ζ32 ζ3 ζ6 ζ32 ζ6 ζ65 ζ65 ζ3 complex lifted from C3×S4 ρ22 3 -1 -1 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1 -1 1 ζ6 ζ65 -3-3√-3/2 ζ6 -3+3√-3/2 ζ65 ζ65 ζ6 ζ6 ζ65 ζ32 ζ32 ζ6 ζ65 ζ3 ζ3 complex lifted from C3×S4 ρ23 3 -1 3 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1 1 -1 ζ65 ζ6 ζ65 -3+3√-3/2 ζ6 -3-3√-3/2 ζ6 ζ65 ζ65 ζ6 ζ65 ζ3 ζ3 ζ32 ζ6 ζ32 complex lifted from C3×S4 ρ24 3 -1 3 -1 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 -1 1 ζ6 ζ65 ζ6 -3-3√-3/2 ζ65 -3+3√-3/2 ζ65 ζ6 ζ32 ζ3 ζ32 ζ6 ζ6 ζ65 ζ3 ζ65 complex lifted from C3×S4 ρ25 3 -1 3 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1 1 -1 ζ6 ζ65 ζ6 -3-3√-3/2 ζ65 -3+3√-3/2 ζ65 ζ6 ζ6 ζ65 ζ6 ζ32 ζ32 ζ3 ζ65 ζ3 complex lifted from C3×S4 ρ26 3 -1 -1 3 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 1 -1 ζ65 ζ6 -3+3√-3/2 ζ65 -3-3√-3/2 ζ6 ζ6 ζ65 ζ3 ζ32 ζ65 ζ65 ζ3 ζ32 ζ6 ζ6 complex lifted from C3×S4 ρ27 3 -1 -1 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1 -1 1 ζ65 ζ6 -3+3√-3/2 ζ65 -3-3√-3/2 ζ6 ζ6 ζ65 ζ65 ζ6 ζ3 ζ3 ζ65 ζ6 ζ32 ζ32 complex lifted from C3×S4 ρ28 6 -2 -2 -2 2 0 6 6 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C22⋊S4 ρ29 6 -2 -2 -2 2 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 1-√-3 1+√-3 1-√-3 1-√-3 1+√-3 1+√-3 -1-√-3 -1+√-3 0 0 0 0 0 0 0 0 complex faithful ρ30 6 -2 -2 -2 2 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 1+√-3 1-√-3 1+√-3 1+√-3 1-√-3 1-√-3 -1+√-3 -1-√-3 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×C22⋊S4
On 24 points - transitive group 24T700
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 11)(2 12)(3 10)(4 24)(5 22)(6 23)(7 14)(8 15)(9 13)(16 21)(17 19)(18 20)
(1 18)(2 16)(3 17)(4 15)(5 13)(6 14)(7 23)(8 24)(9 22)(10 19)(11 20)(12 21)
(1 20)(2 21)(3 19)(4 15)(5 13)(6 14)(7 23)(8 24)(9 22)(10 17)(11 18)(12 16)
(1 18)(2 16)(3 17)(4 24)(5 22)(6 23)(7 14)(8 15)(9 13)(10 19)(11 20)(12 21)
(1 3 2)(4 22 14)(5 23 15)(6 24 13)(7 8 9)(10 21 18)(11 19 16)(12 20 17)
(1 7)(2 8)(3 9)(4 21)(5 19)(6 20)(10 22)(11 23)(12 24)(13 17)(14 18)(15 16)

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

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

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

G:=TransitiveGroup(24,700);

On 24 points - transitive group 24T701
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 18)(2 16)(3 17)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 15)(11 13)(12 14)
(1 15)(2 13)(3 14)(4 8)(5 9)(6 7)(10 18)(11 16)(12 17)(19 24)(20 22)(21 23)
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)(13 16)(14 17)(15 18)(19 24)(20 22)(21 23)
(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)
(7 19 24)(8 20 22)(9 21 23)(10 15 18)(11 13 16)(12 14 17)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)

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

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

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

G:=TransitiveGroup(24,701);

Matrix representation of C3×C22⋊S4 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 12 0 0 0 1 0 12 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 12 12 12 0 0 0 1 0 0
,
 0 12 1 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 12 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 12 12 12 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 12 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 12 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 12 0 0 0 0 1 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 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,1,12,0],[0,0,1,0,0,0,12,12,12,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,1,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,1,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,1,0,0,0,12,1,0],[1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×C22⋊S4 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes S_4
% in TeX

G:=Group("C3xC2^2:S4");
// GroupNames label

G:=SmallGroup(288,1035);
// by ID

G=gap.SmallGroup(288,1035);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,254,1011,185,634,333,6053,1531,3534,608]);
// Polycyclic

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