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G = (C22×S3)⋊A4  order 288 = 25·32

The semidirect product of C22×S3 and A4 acting faithfully

metabelian, soluble, monomial

Aliases: (C22×S3)⋊A4, C24⋊4S3⋊C3, C22⋊A4⋊3S3, (C23×C6)⋊3C6, C24⋊6(C3×S3), C3⋊(C24⋊C6), C22.4(S3×A4), (C2×C6).4(C2×A4), (C3×C22⋊A4)⋊3C2, SmallGroup(288,411)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23×C6 — (C22×S3)⋊A4
Chief series
C1 — C3 — C2×C6 — C23×C6 — C3×C22⋊A4 — (C22×S3)⋊A4
Lower central
C23×C6 — (C22×S3)⋊A4
Upper central
C1

Generators and relations for (C22×S3)⋊A4
 G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e2=f2=g3=1, gbg-1=ab=ba, ac=ca, fdf=gdg-1=ad=da, ae=ea, af=fa, gag-1=b, bc=cb, ede=bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, cg=gc, geg-1=ef=fe, gfg-1=e >

Subgroups: 522 in 77 conjugacy classes, 11 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, A4, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C2×Dic3, C3⋊D4, C2×A4, C22×S3, C22×C6, C22≀C2, C3×A4, C6.D4, C2×C3⋊D4, C22⋊A4, C22⋊A4, C23×C6, S3×A4, C24⋊C6, C24⋊4S3, C3×C22⋊A4, (C22×S3)⋊A4
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, S3×A4, C24⋊C6, (C22×S3)⋊A4

Character table of (C22×S3)⋊A4

 class 12A2B2C2D3A3B3C3D3E46A6B6C6D6E6F6G
 size 13661221616323236666664848
ρ1111111111111111111    trivial
ρ21111-111111-111111-1-1    linear of order 2
ρ31111-11ζ32ζ3ζ3ζ32-111111ζ65ζ6    linear of order 6
ρ41111-11ζ3ζ32ζ32ζ3-111111ζ6ζ65    linear of order 6
ρ5111111ζ3ζ32ζ32ζ3111111ζ32ζ3    linear of order 3
ρ6111111ζ32ζ3ζ3ζ32111111ζ3ζ32    linear of order 3
ρ722220-122-1-10-1-1-1-1-100    orthogonal lifted from S3
ρ822220-1-1-√-3-1+√-3ζ65ζ60-1-1-1-1-100    complex lifted from C3×S3
ρ922220-1-1+√-3-1-√-3ζ6ζ650-1-1-1-1-100    complex lifted from C3×S3
ρ1033-1-1-3300001-1-1-1-1300    orthogonal lifted from C2×A4
ρ1133-1-1330000-1-1-1-1-1300    orthogonal lifted from A4
ρ126-2-22060000022-2-2-200    orthogonal lifted from C24⋊C6
ρ1366-2-20-3000001111-300    orthogonal lifted from S3×A4
ρ146-22-20600000-2-222-200    orthogonal lifted from C24⋊C6
ρ156-22-20-30000011-1-2√-3-1+2√-3100    complex faithful
ρ166-22-20-30000011-1+2√-3-1-2√-3100    complex faithful
ρ176-2-220-300000-1+2√-3-1-2√-311100    complex faithful
ρ186-2-220-300000-1-2√-3-1+2√-311100    complex faithful

Permutation representations of (C22×S3)⋊A4
►On 24 points - transitive group 24T696
Generators in S24
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)

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

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

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

(1 2 3)(4 11 9)(5 12 7)(6 10 8)(13 15 14)(16 24 20)(17 22 21)(18 23 19)
 
G:=sub<Sym(24)| (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,16)(2,18)(3,17)(4,14)(5,13)(6,15)(7,23)(8,22)(9,24)(10,20)(11,19)(12,21), (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,11,9)(5,12,7)(6,10,8)(13,15,14)(16,24,20)(17,22,21)(18,23,19)>;
 
G:=Group( (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,16)(2,18)(3,17)(4,14)(5,13)(6,15)(7,23)(8,22)(9,24)(10,20)(11,19)(12,21), (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,11,9)(5,12,7)(6,10,8)(13,15,14)(16,24,20)(17,22,21)(18,23,19) );
 
G=PermutationGroup([[(1,11),(2,12),(3,10),(4,7),(5,8),(6,9),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,5),(2,6),(3,4),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,16),(2,18),(3,17),(4,14),(5,13),(6,15),(7,23),(8,22),(9,24),(10,20),(11,19),(12,21)], [(1,11),(2,12),(3,10),(4,7),(5,8),(6,9),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,11,9),(5,12,7),(6,10,8),(13,15,14),(16,24,20),(17,22,21),(18,23,19)]])
 
G:=TransitiveGroup(24,696);
 

►On 24 points - transitive group 24T697
Generators in S24
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)

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

(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)

(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)

(4 10 7)(5 11 8)(6 12 9)(16 22 19)(17 23 20)(18 24 21)
 
G:=sub<Sym(24)| (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,16)(2,18)(3,17)(4,14)(5,13)(6,15)(7,23)(8,22)(9,24)(10,20)(11,19)(12,21), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (4,10,7)(5,11,8)(6,12,9)(16,22,19)(17,23,20)(18,24,21)>;
 
G:=Group( (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,16)(2,18)(3,17)(4,14)(5,13)(6,15)(7,23)(8,22)(9,24)(10,20)(11,19)(12,21), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (4,10,7)(5,11,8)(6,12,9)(16,22,19)(17,23,20)(18,24,21) );
 
G=PermutationGroup([[(1,11),(2,12),(3,10),(4,7),(5,8),(6,9),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,5),(2,6),(3,4),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,16),(2,18),(3,17),(4,14),(5,13),(6,15),(7,23),(8,22),(9,24),(10,20),(11,19),(12,21)], [(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(4,10,7),(5,11,8),(6,12,9),(16,22,19),(17,23,20),(18,24,21)]])
 
G:=TransitiveGroup(24,697);
 

Matrix representation of (C22×S3)⋊A4 ►in GL8(𝔽13)

10000000
01000000
00128000
00218000
006610000
00000128
00000218
000006610
,
10000000
01000000
001200000
0011125000
00701000
000001200
0000011125
00000701
,
30000000
09000000
00100000
00010000
00001000
00000100
00000010
00000001
,
012000000
120000000
000001200
0000011125
00000701
001200000
0011125000
00701000
,
10000000
01000000
00128000
00218000
006610000
0000012115
000000120
00000071
,
10000000
01000000
0012115000
000120000
00071000
000001200
0000011125
00000701
,
10000000
01000000
00010000
0012125000
00001000
00000010
0000012125
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,2,6,0,0,0,0,0,2,1,6,0,0,0,0,0,8,8,10,0,0,0,0,0,0,0,0,1,2,6,0,0,0,0,0,2,1,6,0,0,0,0,0,8,8,10],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1],[3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,2,6,0,0,0,0,0,2,1,6,0,0,0,0,0,8,8,10,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,12,7,0,0,0,0,0,5,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,12,7,0,0,0,0,0,5,0,1,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,5,1] >;
 

(C22×S3)⋊A4 in GAP, Magma, Sage, TeX 

(C_2^2\times S_3)\rtimes A_4
 
% in TeX
 
G:=Group("(C2^2xS3):A4");
 
// GroupNames label
 
G:=SmallGroup(288,411);
 
// by ID
 
G=gap.SmallGroup(288,411);
 
# by ID
 
G:=PCGroup([7,-2,-3,-2,2,-2,2,-3,1640,198,1683,94,851,1524,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^3=d^2=e^2=f^2=g^3=1,g*b*g^-1=a*b=b*a,a*c=c*a,f*d*f=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,g*a*g^-1=b,b*c=c*b,e*d*e=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,c*g=g*c,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
 
// generators/relations
 

Export

Character table of (C22×S3)⋊A4 in TeX

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