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G = C4×S3×A4  order 288 = 25·32

Direct product of C4, S3 and A4

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

Aliases: C4×S3×A4, C12⋊2(C2×A4), (C12×A4)⋊6C2, D6.3(C2×A4), (S3×C23).C6, C22⋊2(S3×C12), (C22×C12)⋊2C6, (Dic3×A4)⋊5C2, Dic3⋊2(C2×A4), (C2×A4).14D6, C6.2(C22×A4), (C22×S3)⋊2C12, C23.17(S3×C6), (C6×A4).19C22, (C22×Dic3)⋊2C6, C3⋊1(C2×C4×A4), (S3×C22×C4)⋊C3, C2.1(C2×S3×A4), (C2×S3×A4).2C2, (C3×A4)⋊6(C2×C4), (C2×C6)⋊1(C2×C12), (C22×C4)⋊4(C3×S3), (C22×C6).2(C2×C6), SmallGroup(288,919)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C4×S3×A4
Chief series
C1 — C3 — C2×C6 — C22×C6 — C6×A4 — C2×S3×A4 — C4×S3×A4
Lower central
C2×C6 — C4×S3×A4
Upper central
C1 — C4

Generators and relations for C4×S3×A4
 G = < a,b,c,d,e,f | a4=b3=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 578 in 138 conjugacy classes, 33 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, C23, C23, C32, Dic3, Dic3, C12, C12, A4, A4, D6, D6, C2×C6, C2×C6, C22×C4, C22×C4, C24, C3×S3, C3×C6, C4×S3, C4×S3, C2×Dic3, C2×C12, C2×A4, C2×A4, C22×S3, C22×S3, C22×C6, C23×C4, C3×Dic3, C3×C12, C3×A4, S3×C6, C4×A4, C4×A4, S3×C2×C4, C22×Dic3, C22×C12, C22×A4, S3×C23, S3×C12, S3×A4, C6×A4, C2×C4×A4, S3×C22×C4, Dic3×A4, C12×A4, C2×S3×A4, C4×S3×A4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, A4, D6, C2×C6, C3×S3, C4×S3, C2×C12, C2×A4, S3×C6, C4×A4, C22×A4, S3×C12, S3×A4, C2×C4×A4, C2×S3×A4, C4×S3×A4

Smallest permutation representation of C4×S3×A4
►On 36 points
Generators in S36
(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 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 22 27)(18 23 28)(19 24 25)(20 21 26)

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

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(29 31)(30 32)(33 35)(34 36)

(1 3)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G6H6I6J6K12A12B12C12D12E12F12G12H12I12J12K12L12M12N12O12P
order1222222233333444444446666666666612121212121212121212121212121212
size11333399244881133339924466881212121222444466888812121212

48 irreducible representations

dim111111111122222233333666
type++++++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6C3×S3C4×S3S3×C6S3×C12A4C2×A4C2×A4C2×A4C4×A4S3×A4C2×S3×A4C4×S3×A4
kernelC4×S3×A4Dic3×A4C12×A4C2×S3×A4S3×C22×C4S3×A4C22×Dic3C22×C12S3×C23C22×S3C4×A4C2×A4C22×C4A4C23C22C4×S3Dic3C12D6S3C4C2C1
# reps111124222811222411114112

Matrix representation of C4×S3×A4 ►in GL5(𝔽13)

50000
05000
00100
00010
00001
,
1212000
10000
00100
00010
00001
,
01000
10000
001200
000120
000012
,
10000
01000
00100
0010120
009012
,
10000
01000
001200
00310
000012
,
30000
03000
00308
00001
000410

G:=sub<GL(5,GF(13))| [5,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,10,9,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,3,0,0,0,0,1,0,0,0,0,0,12],[3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,8,1,10] >;
 

C4×S3×A4 in GAP, Magma, Sage, TeX 

C_4\times S_3\times A_4
 
% in TeX
 
G:=Group("C4xS3xA4");
 
// GroupNames label
 
G:=SmallGroup(288,919);
 
// by ID
 
G=gap.SmallGroup(288,919);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,92,648,271,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=b^3=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,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
 

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