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G = C4.S3≀C2  order 288 = 25·32

1st non-split extension by C4 of S3≀C2 acting via S3≀C2/S32=C2

non-abelian, soluble, monomial

Aliases: C4.9S3≀C2, (C3×C12).1D4, C3⋊Dic3.2D4, C32⋊(C4.D4), D6⋊D6.1C2, C12.31D6⋊7C2, C32⋊M4(2)⋊1C2, (C2×S32).C4, C2.3(S32⋊C4), (C4×C3⋊S3).1C22, (C3×C6).2(C22⋊C4), (C2×C3⋊S3).6(C2×C4), SmallGroup(288,375)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C3⋊S3 — C4.S3≀C2
Chief series
C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — D6⋊D6 — C4.S3≀C2
Lower central
C32 — C3×C6 — C2×C3⋊S3 — C4.S3≀C2
Upper central
C1 — C2 — C4

Generators and relations for C4.S3≀C2
 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, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, M4(2), C2×D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C4.D4, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C8⋊S3, S3×D4, C3×C3⋊C8, C32⋊2C8, D6⋊S3, C3×D12, C4×C3⋊S3, C2×S32, C12.31D6, C32⋊M4(2), D6⋊D6, C4.S3≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C4.D4, S3≀C2, S32⋊C4, C4.S3≀C2

Character table of C4.S3≀C2

 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 S3≀C2
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 S3≀C2
ρ15440001-2401-200220011-2-1-1-1-1    orthogonal lifted from S3≀C2
ρ1644-220-21-40-21-11000022-10000    orthogonal lifted from S32⋊C4
ρ17440001-2401-200-2-20011-21111    orthogonal lifted from S3≀C2
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.S3≀C2
►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.S3≀C2 ►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.S3≀C2 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.S3≀C2 in TeX

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