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G = C3⋊S3.2D8order 288 = 25·32

1st non-split extension by C3⋊S3 of D8 acting via D8/C4=C22

non-abelian, soluble, monomial

Aliases: C3⋊S3.2D8, C4.11S3≀C2, (C3×C12).3D4, D6⋊S31C4, C3⋊S3.2SD16, D6⋊D6.2C2, C12.29D67C2, C322(D4⋊C4), C2.5(S32⋊C4), (C2×C3⋊S3).5D4, C4⋊(C32⋊C4)⋊1C2, (C4×C3⋊S3).3C22, C3⋊Dic3.5(C2×C4), (C3×C6).4(C22⋊C4), SmallGroup(288,377)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C3⋊S3.2D8
C1C32C3×C6C3⋊Dic3C4×C3⋊S3D6⋊D6 — C3⋊S3.2D8
C32C3×C6C3⋊Dic3 — C3⋊S3.2D8
C1C2C4

Generators and relations for C3⋊S3.2D8
 G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=c, ab=ba, cac=dbd-1=a-1, dad-1=eae-1=cbc=b-1, ebe-1=a, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 560 in 88 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×2], C22 [×5], S3 [×6], C6 [×4], C8, C2×C4 [×2], D4 [×3], C23, C32, Dic3 [×2], C12 [×2], D6 [×8], C2×C6 [×2], C4⋊C4, C2×C8, C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8, C24, C4×S3 [×2], D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], D4⋊C4, C3⋊Dic3, C3×C12, C32⋊C4, S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×C8, S3×D4, C3×C3⋊C8, D6⋊S3 [×2], C3×D12, C4×C3⋊S3, C2×C32⋊C4, C2×S32, C12.29D6, C4⋊(C32⋊C4), D6⋊D6, C3⋊S3.2D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, D4⋊C4, S3≀C2, S32⋊C4, C3⋊S3.2D8

Character table of C3⋊S3.2D8

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D8A8B8C8D12A12B12C24A24B24C24D
 size 11991212442183636442424666644812121212
ρ1111111111111111111111111111    trivial
ρ21111111111-1-11111-1-1-1-1111-1-1-1-1    linear of order 2
ρ31111-1-111111111-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ41111-1-11111-1-111-1-111111111111    linear of order 2
ρ511-1-11-111-11-ii111-1i-ii-i-1-1-1-i-iii    linear of order 4
ρ611-1-1-1111-11-ii11-11-ii-ii-1-1-1ii-i-i    linear of order 4
ρ711-1-11-111-11i-i111-1-ii-ii-1-1-1ii-i-i    linear of order 4
ρ811-1-1-1111-11i-i11-11i-ii-i-1-1-1-i-iii    linear of order 4
ρ922-2-200222-200220000002220000    orthogonal lifted from D4
ρ1022220022-2-20022000000-2-2-20000    orthogonal lifted from D4
ρ112-22-200220000-2-2002-2-22000-22-22    orthogonal lifted from D8
ρ122-22-200220000-2-200-222-20002-22-2    orthogonal lifted from D8
ρ132-2-2200220000-2-200-2-2--2--2000-2--2--2-2    complex lifted from SD16
ρ142-2-2200220000-2-200--2--2-2-2000--2-2-2--2    complex lifted from SD16
ρ154400001-24000-2100-2-2-2-211-21111    orthogonal lifted from S3≀C2
ρ1644002-2-21-40001-2-11000022-10000    orthogonal lifted from S32⋊C4
ρ17440022-2140001-2-1-10000-2-210000    orthogonal lifted from S3≀C2
ρ184400-2-2-2140001-2110000-2-210000    orthogonal lifted from S3≀C2
ρ194400001-24000-2100222211-2-1-1-1-1    orthogonal lifted from S3≀C2
ρ204400-22-21-40001-21-1000022-10000    orthogonal lifted from S32⋊C4
ρ214400001-2-4000-2100-2i2i-2i2i-1-12-i-iii    complex lifted from S32⋊C4
ρ224400001-2-4000-21002i-2i2i-2i-1-12ii-i-i    complex lifted from S32⋊C4
ρ234-400001-200002-10087858383i-3i0ζ8ζ85ζ87ζ83    complex faithful
ρ244-400001-200002-1008838587-3i3i0ζ87ζ83ζ8ζ85    complex faithful
ρ254-400001-200002-10083887853i-3i0ζ85ζ8ζ83ζ87    complex faithful
ρ264-400001-200002-1008587883-3i3i0ζ83ζ87ζ85ζ8    complex faithful
ρ278-80000-420000-240000000000000    orthogonal faithful

Permutation representations of C3⋊S3.2D8
On 24 points - transitive group 24T666
Generators in S24
(2 22 10)(4 24 12)(6 18 14)(8 20 16)
(1 9 21)(3 11 23)(5 13 17)(7 15 19)
(1 5)(2 6)(3 7)(4 8)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 4 5 8)(2 7 6 3)(9 12 17 20)(10 19 18 11)(13 16 21 24)(14 23 22 15)

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

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

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

G:=TransitiveGroup(24,666);

On 24 points - transitive group 24T667
Generators in S24
(1 21 9)(3 23 11)(5 17 13)(7 19 15)
(2 10 22)(4 12 24)(6 14 18)(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 21 16)(10 15 22 19)(11 18 23 14)(12 13 24 17)

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

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

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

G:=TransitiveGroup(24,667);

Matrix representation of C3⋊S3.2D8 in GL6(𝔽73)

100000
010000
001000
000100
000001
00007272
,
100000
010000
00727200
001000
000010
000001
,
7200000
0720000
001000
00727200
000010
00007272
,
660000
6760000
000010
000001
001000
000100
,
660000
6670000
0000720
000011
0072000
0007200

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[6,67,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,1,0,0,0,0,0,1,0,0] >;

C3⋊S3.2D8 in GAP, Magma, Sage, TeX

C_3\rtimes S_3._2D_8
% in TeX

G:=Group("C3:S3.2D8");
// GroupNames label

G:=SmallGroup(288,377);
// by ID

G=gap.SmallGroup(288,377);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,422,100,675,80,2693,2028,691,797,2372]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=c,a*b=b*a,c*a*c=d*b*d^-1=a^-1,d*a*d^-1=e*a*e^-1=c*b*c=b^-1,e*b*e^-1=a,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

Export

Character table of C3⋊S3.2D8 in TeX

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