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G = C3⋊S3⋊D8order 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⋊D81C2, C321(C2×D8), D6⋊D67C2, (C3×C12).12D4, D6⋊S31C22, C322C83C22, C3⋊Dic3.3C23, C2.9(C2×S3≀C2), C3⋊S33C82C2, (C3×C6).6(C2×D4), (C2×C3⋊S3).30D4, (C4×C3⋊S3).31C22, SmallGroup(288,873)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C3⋊S3⋊D8
C1C32C3×C6C3⋊Dic3D6⋊S3C32⋊D8 — C3⋊S3⋊D8
C32C3×C6C3⋊Dic3 — C3⋊S3⋊D8
C1C2C4

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 [×6], C3 [×2], C4, C4, C22 [×9], S3 [×8], C6 [×6], C8 [×2], C2×C4, D4 [×6], C23 [×2], C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×4], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], C2×D8, C3⋊Dic3, C3×C12, S32 [×4], S3×C6 [×4], C2×C3⋊S3, S3×D4 [×2], C322C8 [×2], D6⋊S3 [×4], C3×D12 [×2], C4×C3⋊S3, C2×S32 [×2], C32⋊D8 [×4], C3⋊S33C8, D6⋊D6 [×2], C3⋊S3⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], 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
ρ112-22-200002200-2-200002-22-200    orthogonal lifted from D8
ρ122-2-2200002200-2-20000-2-22200    orthogonal lifted from D8
ρ132-22-200002200-2-20000-22-2200    orthogonal lifted from D8
ρ142-2-2200002200-2-2000022-2-200    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 11 21)(2 12 22)(3 23 13)(4 24 14)(5 15 17)(6 16 18)(7 19 9)(8 20 10)
(1 11 21)(2 22 12)(3 23 13)(4 14 24)(5 15 17)(6 18 16)(7 19 9)(8 10 20)
(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 24)(16 23)

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

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

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

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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