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G = C4⋊S3≀C2order 288 = 25·32

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

non-abelian, soluble, monomial, rational

Aliases: C41S3≀C2, (C3×C12)⋊2D4, C32⋊C41D4, C3⋊Dic36D4, C32⋊(C41D4), D6⋊D69C2, (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

C1C32C2×C3⋊S3 — C4⋊S3≀C2
C1C32C3⋊S3C2×C3⋊S3C2×S32C2×S3≀C2 — C4⋊S3≀C2
C32C2×C3⋊S3 — C4⋊S3≀C2
C1C2C4

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 [×6], C3 [×2], C4, C4 [×5], C22 [×13], S3 [×8], C6 [×6], C2×C4 [×3], D4 [×12], C23 [×4], C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×4], C42, C2×D4 [×6], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], C41D4, C3⋊Dic3, C3×C12, C32⋊C4 [×4], S32 [×8], S3×C6 [×4], C2×C3⋊S3, S3×D4 [×2], D6⋊S3 [×2], C3×D12 [×2], C4×C3⋊S3, S3≀C2 [×8], C2×C32⋊C4 [×2], C2×S32 [×4], C4×C32⋊C4, D6⋊D6 [×2], C2×S3≀C2 [×4], C4⋊S3≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4, 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 23)(2 16 6 24)(3 21 11 13)(4 14)(5 19 9 15)(7 17)(8 22 12 18)(10 20)
(1 23)(2 18)(3 13)(4 20)(5 15)(6 22)(7 17)(8 24)(9 19)(10 14)(11 21)(12 16)

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

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

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

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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