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G = C4⋊S3≀C2  order 288 = 25·32

The semidirect product of C4 and S3≀C2 acting via S3≀C2/C32⋊C4=C2

non-abelian, soluble, monomial, rational

Aliases: C4⋊1S3≀C2, (C3×C12)⋊2D4, C32⋊C4⋊1D4, C3⋊Dic3⋊6D4, C32⋊(C4⋊1D4), D6⋊D6⋊9C2, (C2×S3≀C2)⋊2C2, (C4×C32⋊C4)⋊4C2, C3⋊S3.3(C2×D4), C2.14(C2×S3≀C2), (C2×S32).2C22, (C3×C6).12(C2×D4), (C2×C3⋊S3).6C23, (C4×C3⋊S3).36C22, (C2×C32⋊C4).15C22, SmallGroup(288,879)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C3⋊S3 — C4⋊S3≀C2
Chief series
C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — C2×S3≀C2 — C4⋊S3≀C2
Lower central
C32 — C2×C3⋊S3 — C4⋊S3≀C2
Upper central
C1 — C2 — C4

Generators and relations for C4⋊S3≀C2
 G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, cac-1=b4, dad=a-1, cbc-1=a-1b9, dbd=b7, dcd=c-1 >

Subgroups: 1048 in 162 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C42, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C4⋊1D4, C3⋊Dic3, C3×C12, C32⋊C4, S32, S3×C6, C2×C3⋊S3, S3×D4, D6⋊S3, C3×D12, C4×C3⋊S3, S3≀C2, C2×C32⋊C4, C2×S32, C4×C32⋊C4, D6⋊D6, C2×S3≀C2, C4⋊S3≀C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C4⋊1D4, S3≀C2, C2×S3≀C2, C4⋊S3≀C2

Character table of C4⋊S3≀C2

 class 12A2B2C2D2E2F2G3A3B4A4B4C4D4E4F6A6B6C6D6E6F12A12B
 size 1199121212124421818181818442424242488
ρ1111111111111111111111111    trivial
ρ211111-11-111-1-111-1-111-111-1-1-1    linear of order 2
ρ3111111-1-11111-1-1-1-1111-11-111    linear of order 2
ρ411111-1-1111-1-1-1-11111-1-111-1-1    linear of order 2
ρ51111-111-111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ61111-1-1111111-1-1-1-111-11-1111    linear of order 2
ρ71111-11-1111-1-111-1-1111-1-11-1-1    linear of order 2
ρ81111-1-1-1-11111111111-1-1-1-111    linear of order 2
ρ922-2-20000222-2000022000022    orthogonal lifted from D4
ρ102-2-220000220000-22-2-2000000    orthogonal lifted from D4
ρ112-2-2200002200002-2-2-2000000    orthogonal lifted from D4
ρ1222-2-2000022-220000220000-2-2    orthogonal lifted from D4
ρ132-22-2000022002-200-2-2000000    orthogonal lifted from D4
ρ142-22-200002200-2200-2-2000000    orthogonal lifted from D4
ρ154400002-21-2-400000-210-1012-1    orthogonal lifted from C2×S3≀C2
ρ16440000221-2400000-210-10-1-21    orthogonal lifted from S3≀C2
ρ17440000-221-2-400000-21010-12-1    orthogonal lifted from C2×S3≀C2
ρ1844002-200-21-4000001-210-10-12    orthogonal lifted from C2×S3≀C2
ρ1944002200-214000001-2-10-101-2    orthogonal lifted from S3≀C2
ρ20440000-2-21-2400000-210101-21    orthogonal lifted from S3≀C2
ρ214400-2-200-214000001-210101-2    orthogonal lifted from S3≀C2
ρ224400-2200-21-4000001-2-1010-12    orthogonal lifted from C2×S3≀C2
ρ238-8000000-42000000-24000000    orthogonal faithful
ρ248-80000002-40000004-2000000    orthogonal faithful

Permutation representations of C4⋊S3≀C2
►On 24 points - transitive group 24T653
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)

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

(1 18)(2 13)(3 20)(4 15)(5 22)(6 17)(7 24)(8 19)(9 14)(10 21)(11 16)(12 23)
 
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,23,6,19)(3,16,11,20)(4,21)(5,14,9,22)(7,24)(8,17,12,13)(10,15), (1,18)(2,13)(3,20)(4,15)(5,22)(6,17)(7,24)(8,19)(9,14)(10,21)(11,16)(12,23)>;
 
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,23,6,19)(3,16,11,20)(4,21)(5,14,9,22)(7,24)(8,17,12,13)(10,15), (1,18)(2,13)(3,20)(4,15)(5,22)(6,17)(7,24)(8,19)(9,14)(10,21)(11,16)(12,23) );
 
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,18),(2,23,6,19),(3,16,11,20),(4,21),(5,14,9,22),(7,24),(8,17,12,13),(10,15)], [(1,18),(2,13),(3,20),(4,15),(5,22),(6,17),(7,24),(8,19),(9,14),(10,21),(11,16),(12,23)]])
 
G:=TransitiveGroup(24,653);
 

Matrix representation of C4⋊S3≀C2 ►in GL6(ℤ)

100000
010000
00-1100
00-1000
0001-1-1
00-1010
,
010000
-100000
00-1100
00-1000
00-1001
0001-1-1
,
100000
010000
0000-11
0011-2-1
0000-10
0001-10
,
0-10000
-100000
0000-11
0011-2-1
0000-10
0010-10

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,-1,0,0,1,0,1,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,1,0,0,1,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,-1,-2,-1,-1,0,0,1,-1,0,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,-1,-2,-1,-1,0,0,1,-1,0,0] >;
 

C4⋊S3≀C2 in GAP, Magma, Sage, TeX 

C_4\rtimes S_3\wr C_2
 
% in TeX
 
G:=Group("C4:S3wrC2");
 
// GroupNames label
 
G:=SmallGroup(288,879);
 
// by ID
 
G=gap.SmallGroup(288,879);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,100,2693,2028,362,797,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,c*a*c^-1=b^4,d*a*d=a^-1,c*b*c^-1=a^-1*b^9,d*b*d=b^7,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C4⋊S3≀C2 in TeX

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