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G = C4.S3wrC2  order 288 = 25·32

1st non-split extension by C4 of S3wrC2 acting via S3wrC2/S32=C2

non-abelian, soluble, monomial

Aliases: C4.9S3wrC2, (C3xC12).1D4, C3:Dic3.2D4, C32:(C4.D4), D6:D6.1C2, C12.31D6:7C2, C32:M4(2):1C2, (C2xS32).C4, C2.3(S32:C4), (C4xC3:S3).1C22, (C3xC6).2(C22:C4), (C2xC3:S3).6(C2xC4), SmallGroup(288,375)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2xC3:S3 — C4.S3wrC2
Chief series
C1 — C32 — C3xC6 — C2xC3:S3 — C4xC3:S3 — D6:D6 — C4.S3wrC2
Lower central
C32 — C3xC6 — C2xC3:S3 — C4.S3wrC2
Upper central
C1 — C2 — C4

Generators and relations for C4.S3wrC2
 G = < a,b,c,d,e | a4=b3=c3=e2=1, d4=a2, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=a-1d3 >

Subgroups: 520 in 83 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2xC4, D4, C23, C32, Dic3, C12, D6, C2xC6, M4(2), C2xD4, C3xS3, C3:S3, C3xC6, C3:C8, C24, C4xS3, D12, C3:D4, C3xD4, C22xS3, C4.D4, C3:Dic3, C3xC12, S32, S3xC6, C2xC3:S3, C8:S3, S3xD4, C3xC3:C8, C32:2C8, D6:S3, C3xD12, C4xC3:S3, C2xS32, C12.31D6, C32:M4(2), D6:D6, C4.S3wrC2
Quotients: C1, C2, C4, C22, C2xC4, D4, C22:C4, C4.D4, S3wrC2, S32:C4, C4.S3wrC2

Character table of C4.S3wrC2

 class 12A2B2C2D3A3B4A4B6A6B6C6D8A8B8C8D12A12B12C24A24B24C24D
 size 11121218442184424241212363644812121212
ρ1111111111111111111111111    trivial
ρ21111111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ311-1-11111111-1-111-1-11111111    linear of order 2
ρ411-1-11111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ5111-1111-1-111-11-ii-ii-1-1-1-i-iii    linear of order 4
ρ611-11111-1-1111-1-iii-i-1-1-1-i-iii    linear of order 4
ρ7111-1111-1-111-11i-ii-i-1-1-1ii-i-i    linear of order 4
ρ811-11111-1-1111-1i-i-ii-1-1-1ii-i-i    linear of order 4
ρ92200-222-2222000000-2-2-20000    orthogonal lifted from D4
ρ102200-2222-2220000002220000    orthogonal lifted from D4
ρ1144-2-20-2140-21110000-2-210000    orthogonal lifted from S3wrC2
ρ124-40004400-4-40000000000000    orthogonal lifted from C4.D4
ρ13442-20-21-40-211-1000022-10000    orthogonal lifted from S32:C4
ρ1444220-2140-21-1-10000-2-210000    orthogonal lifted from S3wrC2
ρ15440001-2401-200220011-2-1-1-1-1    orthogonal lifted from S3wrC2
ρ1644-220-21-40-21-11000022-10000    orthogonal lifted from S32:C4
ρ17440001-2401-200-2-20011-21111    orthogonal lifted from S3wrC2
ρ18440001-2-401-200-2i2i00-1-12ii-i-i    complex lifted from S32:C4
ρ19440001-2-401-2002i-2i00-1-12-i-iii    complex lifted from S32:C4
ρ204-40001-200-120000003i-3i02ζ85ζ3+ζ852ζ8ζ3+ζ82ζ83ζ3+ζ832ζ87ζ3+ζ87    complex faithful
ρ214-40001-200-120000003i-3i02ζ8ζ3+ζ82ζ85ζ3+ζ852ζ87ζ3+ζ872ζ83ζ3+ζ83    complex faithful
ρ224-40001-200-12000000-3i3i02ζ87ζ3+ζ872ζ83ζ3+ζ832ζ8ζ3+ζ82ζ85ζ3+ζ85    complex faithful
ρ234-40001-200-12000000-3i3i02ζ83ζ3+ζ832ζ87ζ3+ζ872ζ85ζ3+ζ852ζ8ζ3+ζ8    complex faithful
ρ248-8000-42004-20000000000000    orthogonal faithful

Permutation representations of C4.S3wrC2
►On 24 points - transitive group 24T662
Generators in S24
(1 7 5 3)(2 4 6 8)(9 18 13 22)(10 23 14 19)(11 20 15 24)(12 17 16 21)

(1 17 10)(3 12 19)(5 21 14)(7 16 23)

(2 11 18)(4 20 13)(6 15 22)(8 24 9)

(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 3)(2 8)(4 6)(5 7)(9 18)(10 12)(11 24)(13 22)(14 16)(15 20)(17 19)(21 23)
 
G:=sub<Sym(24)| (1,7,5,3)(2,4,6,8)(9,18,13,22)(10,23,14,19)(11,20,15,24)(12,17,16,21), (1,17,10)(3,12,19)(5,21,14)(7,16,23), (2,11,18)(4,20,13)(6,15,22)(8,24,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,3)(2,8)(4,6)(5,7)(9,18)(10,12)(11,24)(13,22)(14,16)(15,20)(17,19)(21,23)>;
 
G:=Group( (1,7,5,3)(2,4,6,8)(9,18,13,22)(10,23,14,19)(11,20,15,24)(12,17,16,21), (1,17,10)(3,12,19)(5,21,14)(7,16,23), (2,11,18)(4,20,13)(6,15,22)(8,24,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,3)(2,8)(4,6)(5,7)(9,18)(10,12)(11,24)(13,22)(14,16)(15,20)(17,19)(21,23) );
 
G=PermutationGroup([[(1,7,5,3),(2,4,6,8),(9,18,13,22),(10,23,14,19),(11,20,15,24),(12,17,16,21)], [(1,17,10),(3,12,19),(5,21,14),(7,16,23)], [(2,11,18),(4,20,13),(6,15,22),(8,24,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,3),(2,8),(4,6),(5,7),(9,18),(10,12),(11,24),(13,22),(14,16),(15,20),(17,19),(21,23)]])
 
G:=TransitiveGroup(24,662);
 

Matrix representation of C4.S3wrC2 ►in GL4(F5) generated by

1323
3134
1121
4401
,
0024
0420
0200
1424
,
4400
1000
3012
1413
,
0310
0410
1211
1410
,
1024
0020
0300
0124
G:=sub<GL(4,GF(5))| [1,3,1,4,3,1,1,4,2,3,2,0,3,4,1,1],[0,0,0,1,0,4,2,4,2,2,0,2,4,0,0,4],[4,1,3,1,4,0,0,4,0,0,1,1,0,0,2,3],[0,0,1,1,3,4,2,4,1,1,1,1,0,0,1,0],[1,0,0,0,0,0,3,1,2,2,0,2,4,0,0,4] >;
 

C4.S3wrC2 in GAP, Magma, Sage, TeX 

C_4.S_3\wr C_2
 
% in TeX
 
G:=Group("C4.S3wrC2");
 
// GroupNames label
 
G:=SmallGroup(288,375);
 
// by ID
 
G=gap.SmallGroup(288,375);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,219,675,80,2693,2028,691,797,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^3=c^3=e^2=1,d^4=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=a^-1*d^3>;
 
// generators/relations
 

Export

Character table of C4.S3wrC2 in TeX

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