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G = C33⋊7SD16  order 432 = 24·33

3rd semidirect product of C33 and SD16 acting via SD16/C2=D4

non-abelian, soluble, monomial

Aliases: C33⋊7SD16, C6.17S3≀C2, D6⋊S3.S3, C33⋊4C8⋊3C2, C33⋊5Q8⋊2C2, C3⋊Dic3.12D6, (C32×C6).11D4, C32⋊4(D4.S3), C2.6(C33⋊D4), C3⋊2(C32⋊2SD16), (C3×D6⋊S3).1C2, (C3×C6).17(C3⋊D4), (C3×C3⋊Dic3).9C22, SmallGroup(432,584)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊Dic3 — C33⋊7SD16
Chief series
C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊5Q8 — C33⋊7SD16
Lower central
C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊7SD16
Upper central
C1 — C2

Generators and relations for C33⋊7SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b-1, eae=a-1, bc=cb, dbd-1=a, be=eb, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 508 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×C6, C3×C6, C3⋊C8, Dic6, C3⋊D4, C3×D4, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, C62, D4.S3, S3×C32, C32×C6, C32⋊2C8, D6⋊S3, C32⋊2Q8, C3×C3⋊D4, C3×C3⋊Dic3, C3×C3⋊Dic3, S3×C3×C6, C32⋊2SD16, C33⋊4C8, C3×D6⋊S3, C33⋊5Q8, C33⋊7SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊D4, D4.S3, S3≀C2, C32⋊2SD16, C33⋊D4, C33⋊7SD16

Character table of C33⋊7SD16

 class 12A2B3A3B3C3D3E3F4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N8A8B12A12B12C
 size 1112244448183624444812121212121212125454363636
ρ1111111111111111111111111111111    trivial
ρ211-11111111-1111111-1-1-1-1-1-1-1-111-11-1    linear of order 2
ρ31111111111-111111111111111-1-1-11-1    linear of order 2
ρ411-111111111111111-1-1-1-1-1-1-1-1-1-1111    linear of order 2
ρ5220222222-2022222200000000000-20    orthogonal lifted from D4
ρ6222-1-12-12-120-1-122-1-1-12-1-12-1-1-1000-10    orthogonal lifted from S3
ρ722-2-1-12-12-120-1-122-1-11-211-2111000-10    orthogonal lifted from D6
ρ8220-1-12-12-1-20-1-122-1-1-√-30√-3√-30√-3-√-3-√-300010    complex lifted from C3⋊D4
ρ9220-1-12-12-1-20-1-122-1-1√-30-√-3-√-30-√-3√-3√-300010    complex lifted from C3⋊D4
ρ102-2022222200-2-2-2-2-2-200000000-√-2√-2000    complex lifted from SD16
ρ112-2022222200-2-2-2-2-2-200000000√-2-√-2000    complex lifted from SD16
ρ1244241-211-20041-211-2-1-1-12-1-1-1200000    orthogonal lifted from S3≀C2
ρ134404-21-2-210-24-21-2-210000000000101    orthogonal lifted from S3≀C2
ρ144404-21-2-21024-21-2-210000000000-10-1    orthogonal lifted from S3≀C2
ρ1544-241-211-20041-211-2111-2111-200000    orthogonal lifted from S3≀C2
ρ164-40-2-24-24-20022-4-4220000000000000    symplectic lifted from D4.S3, Schur index 2
ρ174-404-21-2-2100-42-122-10000000000-√30√3    symplectic lifted from C32⋊2SD16, Schur index 2
ρ184-404-21-2-2100-42-122-10000000000√30-√3    symplectic lifted from C32⋊2SD16, Schur index 2
ρ194-4041-211-200-4-12-1-12-√-3-√-3-√-30√-3√-3√-3000000    complex lifted from C32⋊2SD16
ρ2044-2-2-1-3√-3/2-2-1+3√-3/21100-2-1+3√-3/2-21-1-3√-3/21ζ31ζ321+√-31ζ32ζ31-√-300000    complex lifted from C33⋊D4
ρ214-40-2-1-3√-3/2-2-1+3√-3/2110021-3√-3/22-11+3√-3/2-13+√-3/2-√-3-3+√-3/20√-33-√-3/2-3-√-3/2000000    complex faithful
ρ22442-2-1-3√-3/2-2-1+3√-3/21100-2-1+3√-3/2-21-1-3√-3/21ζ65-1ζ6-1-√-3-1ζ6ζ65-1+√-300000    complex lifted from C33⋊D4
ρ234-40-2-1-3√-3/2-2-1+3√-3/2110021-3√-3/22-11+3√-3/2-1-3-√-3/2√-33-√-3/20-√-3-3+√-3/23+√-3/2000000    complex faithful
ρ244-40-2-1+3√-3/2-2-1-3√-3/2110021+3√-3/22-11-3√-3/2-1-3+√-3/2-√-33+√-3/20√-3-3-√-3/23-√-3/2000000    complex faithful
ρ2544-2-2-1+3√-3/2-2-1-3√-3/21100-2-1-3√-3/2-21-1+3√-3/21ζ321ζ31-√-31ζ3ζ321+√-300000    complex lifted from C33⋊D4
ρ264-40-2-1+3√-3/2-2-1-3√-3/2110021+3√-3/22-11-3√-3/2-13-√-3/2√-3-3-√-3/20-√-33+√-3/2-3+√-3/2000000    complex faithful
ρ274-4041-211-200-4-12-1-12√-3√-3√-30-√-3-√-3-√-3000000    complex lifted from C32⋊2SD16
ρ28442-2-1+3√-3/2-2-1-3√-3/21100-2-1-3√-3/2-21-1+3√-3/21ζ6-1ζ65-1+√-3-1ζ65ζ6-1-√-300000    complex lifted from C33⋊D4
ρ29880-4222-4-100-422-42-10000000000000    orthogonal lifted from C33⋊D4
ρ308-80-4222-4-1004-2-24-210000000000000    symplectic faithful, Schur index 2

Permutation representations of C33⋊7SD16
►On 24 points - transitive group 24T1291
Generators in S24
(2 23 9)(4 11 17)(6 19 13)(8 15 21)

(1 16 22)(3 24 10)(5 12 18)(7 20 14)

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

(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 11)(10 14)(13 15)(17 23)(19 21)(20 24)
 
G:=sub<Sym(24)| (2,23,9)(4,11,17)(6,19,13)(8,15,21), (1,16,22)(3,24,10)(5,12,18)(7,20,14), (1,22,16)(2,9,23)(3,24,10)(4,11,17)(5,18,12)(6,13,19)(7,20,14)(8,15,21), (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)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)>;
 
G:=Group( (2,23,9)(4,11,17)(6,19,13)(8,15,21), (1,16,22)(3,24,10)(5,12,18)(7,20,14), (1,22,16)(2,9,23)(3,24,10)(4,11,17)(5,18,12)(6,13,19)(7,20,14)(8,15,21), (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)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24) );
 
G=PermutationGroup([[(2,23,9),(4,11,17),(6,19,13),(8,15,21)], [(1,16,22),(3,24,10),(5,12,18),(7,20,14)], [(1,22,16),(2,9,23),(3,24,10),(4,11,17),(5,18,12),(6,13,19),(7,20,14),(8,15,21)], [(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),(9,11),(10,14),(13,15),(17,23),(19,21),(20,24)]])
 
G:=TransitiveGroup(24,1291);
 

Matrix representation of C33⋊7SD16 ►in GL4(𝔽7) generated by

1040
5614
4406
0001
,
5353
3523
0010
0004
,
3632
6342
0020
0004
,
0630
0245
3425
2263
,
1005
5602
4416
0006
G:=sub<GL(4,GF(7))| [1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[0,0,3,2,6,2,4,2,3,4,2,6,0,5,5,3],[1,5,4,0,0,6,4,0,0,0,1,0,5,2,6,6] >;
 

C33⋊7SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_7{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:7SD16");
 
// GroupNames label
 
G:=SmallGroup(432,584);
 
// by ID
 
G=gap.SmallGroup(432,584);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,254,135,58,1684,571,298,677,1027,14118]);
 
// 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=b^-1,e*a*e=a^-1,b*c=c*b,d*b*d^-1=a,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

Export

Character table of C33⋊7SD16 in TeX

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