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G = C4.S3≀C2order 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, C12.31D67C2, 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

C1C32C2×C3⋊S3 — C4.S3≀C2
C1C32C3×C6C2×C3⋊S3C4×C3⋊S3D6⋊D6 — C4.S3≀C2
C32C3×C6C2×C3⋊S3 — C4.S3≀C2
C1C2C4

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 [×3], C3 [×2], C4, C4, C22 [×5], S3 [×5], C6 [×4], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, Dic3 [×2], C12 [×2], D6 [×8], C2×C6 [×2], M4(2) [×2], C2×D4, C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3 [×2], D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C4.D4, C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, C8⋊S3, S3×D4, C3×C3⋊C8, C322C8, D6⋊S3, C3×D12, C4×C3⋊S3, C2×S32 [×2], C12.31D6, C32⋊M4(2), D6⋊D6, C4.S3≀C2
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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-3i085ζ3858ζ3883ζ38387ζ387    complex faithful
ρ214-40001-200-120000003i-3i08ζ3885ζ38587ζ38783ζ383    complex faithful
ρ224-40001-200-12000000-3i3i087ζ38783ζ3838ζ3885ζ385    complex faithful
ρ234-40001-200-12000000-3i3i083ζ38387ζ38785ζ3858ζ38    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

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Character table of C4.S3≀C2 in TeX

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