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G = C3⋊S3⋊D8  order 288 = 25·32

The semidirect product of C3⋊S3 and D8 acting via D8/C4=C22

non-abelian, soluble, monomial

Aliases: C3⋊S3⋊D8, C4.6S3≀C2, C32⋊D8⋊1C2, C32⋊1(C2×D8), D6⋊D6⋊7C2, (C3×C12).12D4, D6⋊S3⋊1C22, C32⋊2C8⋊3C22, C3⋊Dic3.3C23, C2.9(C2×S3≀C2), C3⋊S3⋊3C8⋊2C2, (C3×C6).6(C2×D4), (C2×C3⋊S3).30D4, (C4×C3⋊S3).31C22, SmallGroup(288,873)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊Dic3 — C3⋊S3⋊D8
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8 — C3⋊S3⋊D8
Lower central
C32 — C3×C6 — C3⋊Dic3 — C3⋊S3⋊D8
Upper central
C1 — C2 — C4

Generators and relations for C3⋊S3⋊D8
 G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, cac=dbd-1=a-1, dad-1=b, ae=ea, cbc=ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 848 in 130 conjugacy classes, 25 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, D8, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C2×D8, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×D4, C32⋊2C8, D6⋊S3, C3×D12, C4×C3⋊S3, C2×S32, C32⋊D8, C3⋊S3⋊3C8, D6⋊D6, C3⋊S3⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C2×D8, S3≀C2, C2×S3≀C2, C3⋊S3⋊D8

Character table of C3⋊S3⋊D8

 class 12A2B2C2D2E2F2G3A3B4A4B6A6B6C6D6E6F8A8B8C8D12A12B
 size 1199121212124421844242424241818181888
ρ1111111111111111111111111    trivial
ρ211-1-11-11-111-1111-111-1-111-1-1-1    linear of order 2
ρ31111-1-1-1-1111111-1-1-1-1111111    linear of order 2
ρ411-1-1-11-1111-11111-1-11-111-1-1-1    linear of order 2
ρ511-1-1-111-111-11111-11-11-1-11-1-1    linear of order 2
ρ61111-1-111111111-1-111-1-1-1-111    linear of order 2
ρ711-1-11-1-1111-1111-11-111-1-11-1-1    linear of order 2
ρ8111111-1-111111111-1-1-1-1-1-111    linear of order 2
ρ92222000022-2-22200000000-2-2    orthogonal lifted from D4
ρ1022-2-20000222-2220000000022    orthogonal lifted from D4
ρ112-22-200002200-2-20000√2-√2√2-√200    orthogonal lifted from D8
ρ122-2-2200002200-2-20000-√2-√2√2√200    orthogonal lifted from D8
ρ132-22-200002200-2-20000-√2√2-√2√200    orthogonal lifted from D8
ρ142-2-2200002200-2-20000√2√2-√2-√200    orthogonal lifted from D8
ρ154400-22001-2-401-2-11000000-12    orthogonal lifted from C2×S3≀C2
ρ1644002-2001-2-401-21-1000000-12    orthogonal lifted from C2×S3≀C2
ρ174400002-2-21-40-2100-1100002-1    orthogonal lifted from C2×S3≀C2
ρ1844000022-2140-2100-1-10000-21    orthogonal lifted from S3≀C2
ρ194400-2-2001-2401-2110000001-2    orthogonal lifted from S3≀C2
ρ20440022001-2401-2-1-10000001-2    orthogonal lifted from S3≀C2
ρ21440000-2-2-2140-2100110000-21    orthogonal lifted from S3≀C2
ρ22440000-22-21-40-21001-100002-1    orthogonal lifted from C2×S3≀C2
ρ238-80000002-400-240000000000    orthogonal faithful
ρ248-8000000-42004-20000000000    orthogonal faithful

Permutation representations of C3⋊S3⋊D8
►On 24 points - transitive group 24T659
Generators in S24
(1 9 21)(2 10 22)(3 23 11)(4 24 12)(5 13 17)(6 14 18)(7 19 15)(8 20 16)

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

(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)
 
G:=sub<Sym(24)| (1,9,21)(2,10,22)(3,23,11)(4,24,12)(5,13,17)(6,14,18)(7,19,15)(8,20,16), (1,9,21)(2,22,10)(3,23,11)(4,12,24)(5,13,17)(6,18,14)(7,19,15)(8,16,20), (9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)>;
 
G:=Group( (1,9,21)(2,10,22)(3,23,11)(4,24,12)(5,13,17)(6,14,18)(7,19,15)(8,20,16), (1,9,21)(2,22,10)(3,23,11)(4,12,24)(5,13,17)(6,18,14)(7,19,15)(8,16,20), (9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21) );
 
G=PermutationGroup([[(1,9,21),(2,10,22),(3,23,11),(4,24,12),(5,13,17),(6,14,18),(7,19,15),(8,20,16)], [(1,9,21),(2,22,10),(3,23,11),(4,12,24),(5,13,17),(6,18,14),(7,19,15),(8,16,20)], [(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21)]])
 
G:=TransitiveGroup(24,659);
 

Matrix representation of C3⋊S3⋊D8 ►in GL6(𝔽73)

100000
010000
0072100
0072000
0000721
0000720
,
100000
010000
0007200
0017200
0000721
0000720
,
7200000
0720000
000100
001000
000001
000010
,
57160000
57570000
000010
000001
0007200
0072000
,
57160000
16160000
000010
000001
001000
000100

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,1,0,0],[57,16,0,0,0,0,16,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C3⋊S3⋊D8 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3\rtimes D_8
 
% in TeX
 
G:=Group("C3:S3:D8");
 
// GroupNames label
 
G:=SmallGroup(288,873);
 
// by ID
 
G=gap.SmallGroup(288,873);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,100,675,346,80,2693,2028,362,797,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=e^2=1,a*b=b*a,c*a*c=d*b*d^-1=a^-1,d*a*d^-1=b,a*e=e*a,c*b*c=e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C3⋊S3⋊D8 in TeX

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