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G = C3×C4⋊S4order 288 = 25·32

Direct product of C3 and C4⋊S4

direct product, non-abelian, soluble, monomial

Aliases: C3×C4⋊S4, C123S4, C4⋊(C3×S4), (C6×S4)⋊4C2, (C2×S4)⋊1C6, (C4×A4)⋊1C6, (C3×A4)⋊4D4, A41(C3×D4), (C2×C6)⋊2D12, C2.4(C6×S4), C22⋊(C3×D12), (C12×A4)⋊2C2, C6.41(C2×S4), (C22×C12)⋊4S3, C23.3(S3×C6), (C22×C6).10D6, (C6×A4).10C22, (C2×A4).3(C2×C6), (C22×C4)⋊2(C3×S3), SmallGroup(288,898)

Series: Derived Chief Lower central Upper central

C1C22C2×A4 — C3×C4⋊S4
C1C22A4C2×A4C6×A4C6×S4 — C3×C4⋊S4
A4C2×A4 — C3×C4⋊S4
C1C6C12

Generators and relations for C3×C4⋊S4
 G = < a,b,c,d,e,f | a3=b4=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 466 in 118 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4, C4 [×3], C22, C22 [×8], S3 [×2], C6, C6 [×6], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, C12, C12 [×5], A4, A4, D6 [×2], C2×C6, C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3×C6, D12, C2×C12 [×4], C3×D4 [×6], S4 [×2], C2×A4, C2×A4, C22×C6, C22×C6 [×2], C4⋊D4, C3×C12, C3×A4, S3×C6 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C4×A4, C4×A4, C22×C12, C6×D4 [×3], C2×S4 [×2], C3×D12, C3×S4 [×2], C6×A4, C3×C4⋊D4, C4⋊S4, C12×A4, C6×S4 [×2], C3×C4⋊S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, C3×S3, D12, C3×D4, S4, S3×C6, C2×S4, C3×D12, C3×S4, C4⋊S4, C6×S4, C3×C4⋊S4

Smallest permutation representation of C3×C4⋊S4
On 36 points
Generators in S36
(1 35 23)(2 36 24)(3 33 21)(4 34 22)(5 12 31)(6 9 32)(7 10 29)(8 11 30)(13 19 27)(14 20 28)(15 17 25)(16 18 26)
(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)
(1 3)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(33 35)(34 36)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(25 27)(26 28)(29 31)(30 32)
(1 13 5)(2 14 6)(3 15 7)(4 16 8)(9 36 20)(10 33 17)(11 34 18)(12 35 19)(21 25 29)(22 26 30)(23 27 31)(24 28 32)
(1 4)(2 3)(5 16)(6 15)(7 14)(8 13)(9 17)(10 20)(11 19)(12 18)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I6J6K6L6M12A12B12C12D12E···12J12K12L12M12N
order12222233333444466666666666661212121212···1212121212
size11331212118882612121133338881212121222668···812121212

42 irreducible representations

dim11111122222222333366
type++++++++++
imageC1C2C2C3C6C6S3D4D6C3×S3C3×D4D12S3×C6C3×D12S4C2×S4C3×S4C6×S4C4⋊S4C3×C4⋊S4
kernelC3×C4⋊S4C12×A4C6×S4C4⋊S4C4×A4C2×S4C22×C12C3×A4C22×C6C22×C4A4C2×C6C23C22C12C6C4C2C3C1
# reps11222411122224224412

Matrix representation of C3×C4⋊S4 in GL5(𝔽13)

90000
09000
00100
00010
00001
,
012000
10000
001200
000120
000012
,
10000
01000
001200
001201
001210
,
10000
01000
000112
001012
000012
,
10000
01000
000120
001120
000121
,
01000
10000
00010
00100
00001

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,12,12,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C3×C4⋊S4 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes S_4
% in TeX

G:=Group("C3xC4:S4");
// GroupNames label

G:=SmallGroup(288,898);
// by ID

G=gap.SmallGroup(288,898);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,197,92,1684,6053,285,3534,475]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^4=c^2=d^2=e^3=f^2=1,a*b=b*a,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,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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