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G = C33⋊8SD16  order 432 = 24·33

4th semidirect product of C33 and SD16 acting via SD16/C2=D4

non-abelian, soluble, monomial

Aliases: C33⋊8SD16, C6.10S3≀C2, C32⋊2C8⋊2S3, (C3×C6).11D12, C33⋊5Q8⋊1C2, C3⋊Dic3.15D6, (C32×C6).16D4, C33⋊9D4.1C2, C32⋊4(C24⋊C2), C2.4(C32⋊2D12), C3⋊1(C32⋊2SD16), (C3×C32⋊2C8)⋊2C2, (C3×C3⋊Dic3).2C22, SmallGroup(432,589)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊Dic3 — C33⋊8SD16
Chief series
C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊9D4 — C33⋊8SD16
Lower central
C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊8SD16
Upper central
C1 — C2

Generators and relations for C33⋊8SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=b, bc=cb, dbd-1=a-1, ebe=a, cd=dc, ece=c-1, ede=d3 >

Subgroups: 640 in 84 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C24, Dic6, D12, C3⋊D4, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C24⋊C2, C3×C3⋊S3, C32×C6, C32⋊2C8, D6⋊S3, C3⋊D12, C32⋊2Q8, C3×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C32⋊2SD16, C3×C32⋊2C8, C33⋊9D4, C33⋊5Q8, C33⋊8SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, D12, C24⋊C2, S3≀C2, C32⋊2SD16, C32⋊2D12, C33⋊8SD16

Character table of C33⋊8SD16

 class 12A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B12A12B12C12D24A24B24C24D
 size 113624488183624488363618181818363618181818
ρ1111111111111111111111111111    trivial
ρ211-1111111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ311-1111111-111111-1-11111-1-11111    linear of order 2
ρ4111111111-11111111-1-111-1-1-1-1-1-1    linear of order 2
ρ5220-122-1-120-122-1-100-2-2-1-1001111    orthogonal lifted from D6
ρ622022222-20222220000-2-2000000    orthogonal lifted from D4
ρ7220-122-1-120-122-1-10022-1-100-1-1-1-1    orthogonal lifted from S3
ρ8220-122-1-1-20-122-1-100001100-√3-√3√3√3    orthogonal lifted from D12
ρ9220-122-1-1-20-122-1-100001100√3√3-√3-√3    orthogonal lifted from D12
ρ102-202222200-2-2-2-2-200√-2-√-20000√-2-√-2-√-2√-2    complex lifted from SD16
ρ112-202222200-2-2-2-2-200-√-2√-20000-√-2√-2√-2-√-2    complex lifted from SD16
ρ122-20-122-1-1001-2-21100-√-2√-2√3-√300ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32    complex lifted from C24⋊C2
ρ132-20-122-1-1001-2-21100√-2-√-2√3-√300ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32    complex lifted from C24⋊C2
ρ142-20-122-1-1001-2-21100√-2-√-2-√3√300ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3    complex lifted from C24⋊C2
ρ152-20-122-1-1001-2-21100-√-2√-2-√3√300ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3    complex lifted from C24⋊C2
ρ1644-24-21-21004-21-21110000000000    orthogonal lifted from S3≀C2
ρ174424-21-21004-21-21-1-10000000000    orthogonal lifted from S3≀C2
ρ1844041-21-20-241-21-2000000110000    orthogonal lifted from S3≀C2
ρ1944041-21-20241-21-2000000-1-10000    orthogonal lifted from S3≀C2
ρ204-4041-21-200-4-12-12000000√3-√30000    symplectic lifted from C32⋊2SD16, Schur index 2
ρ214-4041-21-200-4-12-12000000-√3√30000    symplectic lifted from C32⋊2SD16, Schur index 2
ρ224-404-21-2100-42-12-1√-3-√-30000000000    complex lifted from C32⋊2SD16
ρ234-404-21-2100-42-12-1-√-3√-30000000000    complex lifted from C32⋊2SD16
ρ248-80-4-422-10044-2-21000000000000    orthogonal faithful
ρ25880-4-422-100-4-422-1000000000000    orthogonal lifted from C32⋊2D12
ρ26880-42-4-1200-42-4-12000000000000    orthogonal lifted from C32⋊2D12
ρ278-80-42-4-12004-241-2000000000000    symplectic faithful, Schur index 2

Permutation representations of C33⋊8SD16
►On 24 points - transitive group 24T1307
Generators in S24
(1 15 21)(2 22 16)(3 23 9)(4 10 24)(5 11 17)(6 18 12)(7 19 13)(8 14 20)

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

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

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

(2 4)(3 7)(6 8)(9 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)
 
G:=sub<Sym(24)| (1,15,21)(2,22,16)(3,23,9)(4,10,24)(5,11,17)(6,18,12)(7,19,13)(8,14,20), (1,21,15)(2,22,16)(3,9,23)(4,10,24)(5,17,11)(6,18,12)(7,13,19)(8,14,20), (1,21,15)(2,22,16)(3,23,9)(4,24,10)(5,17,11)(6,18,12)(7,19,13)(8,20,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)>;
 
G:=Group( (1,15,21)(2,22,16)(3,23,9)(4,10,24)(5,11,17)(6,18,12)(7,19,13)(8,14,20), (1,21,15)(2,22,16)(3,9,23)(4,10,24)(5,17,11)(6,18,12)(7,13,19)(8,14,20), (1,21,15)(2,22,16)(3,23,9)(4,24,10)(5,17,11)(6,18,12)(7,19,13)(8,20,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24) );
 
G=PermutationGroup([[(1,15,21),(2,22,16),(3,23,9),(4,10,24),(5,11,17),(6,18,12),(7,19,13),(8,14,20)], [(1,21,15),(2,22,16),(3,9,23),(4,10,24),(5,17,11),(6,18,12),(7,13,19),(8,14,20)], [(1,21,15),(2,22,16),(3,23,9),(4,24,10),(5,17,11),(6,18,12),(7,19,13),(8,20,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(9,19),(10,22),(11,17),(12,20),(13,23),(14,18),(15,21),(16,24)]])
 
G:=TransitiveGroup(24,1307);
 

Matrix representation of C33⋊8SD16 ►in GL6(𝔽73)

100000
010000
0000072
0000720
0001720
0010072
,
100000
010000
0000072
0007210
0007200
0010072
,
0720000
1720000
001000
000100
000010
000001
,
36620000
11250000
000010
001000
000001
000100
,
0720000
7200000
001000
000010
000100
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72,72,0,0,0,72,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,1,0,0,0,0,72,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,11,0,0,0,0,62,25,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1] >;
 

C33⋊8SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_8{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:8SD16");
 
// GroupNames label
 
G:=SmallGroup(432,589);
 
// by ID
 
G=gap.SmallGroup(432,589);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,36,254,58,1684,1691,298,677,348,1027,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=b,b*c=c*b,d*b*d^-1=a^-1,e*b*e=a,c*d=d*c,e*c*e=c^-1,e*d*e=d^3>;
 
// generators/relations
 

Export

Character table of C33⋊8SD16 in TeX

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