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G = C33⋊6SD16  order 432 = 24·33

2nd semidirect product of C33 and SD16 acting via SD16/C2=D4

non-abelian, soluble, monomial

Aliases: C33⋊6SD16, C6.16S3≀C2, C33⋊4C8⋊2C2, C32⋊2Q8⋊1S3, C3⋊Dic3.11D6, (C32×C6).10D4, C33⋊9D4.2C2, C2.5(C33⋊D4), C32⋊4(Q8⋊2S3), C3⋊3(C32⋊2SD16), (C3×C32⋊2Q8)⋊1C2, (C3×C6).16(C3⋊D4), (C3×C3⋊Dic3).8C22, SmallGroup(432,583)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊Dic3 — C33⋊6SD16
Chief series
C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊9D4 — C33⋊6SD16
Lower central
C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊6SD16
Upper central
C1 — C2

Generators and relations for C33⋊6SD16
 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, dcd-1=ece=c-1, ede=d3 >

Subgroups: 588 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, C3⋊C8, Dic6, D12, C3⋊D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, Q8⋊2S3, C3×C3⋊S3, C32×C6, C32⋊2C8, D6⋊S3, C3⋊D12, C32⋊2Q8, C3×Dic6, C32×Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C32⋊2SD16, C33⋊4C8, C3×C32⋊2Q8, C33⋊9D4, C33⋊6SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊D4, Q8⋊2S3, S3≀C2, C32⋊2SD16, C33⋊D4, C33⋊6SD16

Character table of C33⋊6SD16

 class 12A2B3A3B3C3D3E3F4A4B6A6B6C6D6E6F6G6H8A8B12A12B12C12D12E12F12G12H12I
 size 1136244448121824444836365454121212121212121236
ρ1111111111111111111111111111111    trivial
ρ211-1111111-11111111-1-111-1-1-1-1-1-1-1-11    linear of order 2
ρ3111111111-1111111111-1-1-1-1-1-1-1-1-1-11    linear of order 2
ρ411-111111111111111-1-1-1-1111111111    linear of order 2
ρ5220-122-1-1-122-12-1-12-10000-12-1-1-1-12-1-1    orthogonal lifted from S3
ρ62202222220-2222222000000000000-2    orthogonal lifted from D4
ρ7220-122-1-1-1-22-12-1-12-100001-21111-21-1    orthogonal lifted from D6
ρ8220-122-1-1-10-2-12-1-12-10000-√-30√-3-√-3√-3-√-30√-31    complex lifted from C3⋊D4
ρ9220-122-1-1-10-2-12-1-12-10000√-30-√-3√-3-√-3√-30-√-31    complex lifted from C3⋊D4
ρ102-2022222200-2-2-2-2-2-200√-2-√-2000000000    complex lifted from SD16
ρ112-2022222200-2-2-2-2-2-200-√-2√-2000000000    complex lifted from SD16
ρ1244241-2-2-21004-2-2-211-1-100000000000    orthogonal lifted from S3≀C2
ρ134404-2111-2-204111-2-200001111-2-2110    orthogonal lifted from S3≀C2
ρ144-40-244-2-2-2002-422-420000000000000    orthogonal lifted from Q8⋊2S3
ρ154404-2111-2204111-2-20000-1-1-1-122-1-10    orthogonal lifted from S3≀C2
ρ1644-241-2-2-21004-2-2-2111100000000000    orthogonal lifted from S3≀C2
ρ174-404-2111-200-4-1-1-1220000√3-√3-√3-√300√3√30    symplectic lifted from C32⋊2SD16, Schur index 2
ρ184-404-2111-200-4-1-1-1220000-√3√3√3√300-√3-√30    symplectic lifted from C32⋊2SD16, Schur index 2
ρ19440-2-21-1+3√-3/2-1-3√-3/21-20-21-1+3√-3/2-1-3√-3/2-210000ζ321ζ3ζ321-√-31+√-31ζ30    complex lifted from C33⋊D4
ρ204-4041-2-2-2100-4222-1-1√-3-√-300000000000    complex lifted from C32⋊2SD16
ρ21440-2-21-1-3√-3/2-1+3√-3/21-20-21-1-3√-3/2-1+3√-3/2-210000ζ31ζ32ζ31+√-31-√-31ζ320    complex lifted from C33⋊D4
ρ224-4041-2-2-2100-4222-1-1-√-3√-300000000000    complex lifted from C32⋊2SD16
ρ23440-2-21-1+3√-3/2-1-3√-3/2120-21-1+3√-3/2-1-3√-3/2-210000ζ6-1ζ65ζ6-1+√-3-1-√-3-1ζ650    complex lifted from C33⋊D4
ρ244-40-2-21-1+3√-3/2-1-3√-3/21002-11-3√-3/21+3√-3/22-10000ζ4ζ32+2ζ4√3ζ4ζ3+2ζ4ζ43ζ32+2ζ4300-√3ζ43ζ3+2ζ430    complex faithful
ρ254-40-2-21-1-3√-3/2-1+3√-3/21002-11+3√-3/21-3√-3/22-10000ζ43ζ3+2ζ43√3ζ43ζ32+2ζ43ζ4ζ3+2ζ400-√3ζ4ζ32+2ζ40    complex faithful
ρ264-40-2-21-1+3√-3/2-1-3√-3/21002-11-3√-3/21+3√-3/22-10000ζ43ζ32+2ζ43-√3ζ43ζ3+2ζ43ζ4ζ32+2ζ400√3ζ4ζ3+2ζ40    complex faithful
ρ27440-2-21-1-3√-3/2-1+3√-3/2120-21-1-3√-3/2-1+3√-3/2-210000ζ65-1ζ6ζ65-1-√-3-1+√-3-1ζ60    complex lifted from C33⋊D4
ρ284-40-2-21-1-3√-3/2-1+3√-3/21002-11+3√-3/21-3√-3/22-10000ζ4ζ3+2ζ4-√3ζ4ζ32+2ζ4ζ43ζ3+2ζ4300√3ζ43ζ32+2ζ430    complex faithful
ρ298-80-42-422-10044-2-2-210000000000000    orthogonal faithful
ρ30880-42-422-100-4-4222-10000000000000    orthogonal lifted from C33⋊D4

Permutation representations of C33⋊6SD16
►On 24 points - transitive group 24T1315
Generators in S24
(1 21 10)(2 11 22)(3 12 23)(4 24 13)(5 17 14)(6 15 18)(7 16 19)(8 20 9)

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

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

(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 18)(10 21)(11 24)(12 19)(13 22)(14 17)(15 20)(16 23)
 
G:=sub<Sym(24)| (1,21,10)(2,11,22)(3,12,23)(4,24,13)(5,17,14)(6,15,18)(7,16,19)(8,20,9), (1,10,21)(2,11,22)(3,23,12)(4,24,13)(5,14,17)(6,15,18)(7,19,16)(8,20,9), (1,10,21)(2,22,11)(3,12,23)(4,24,13)(5,14,17)(6,18,15)(7,16,19)(8,20,9), (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)(9,18)(10,21)(11,24)(12,19)(13,22)(14,17)(15,20)(16,23)>;
 
G:=Group( (1,21,10)(2,11,22)(3,12,23)(4,24,13)(5,17,14)(6,15,18)(7,16,19)(8,20,9), (1,10,21)(2,11,22)(3,23,12)(4,24,13)(5,14,17)(6,15,18)(7,19,16)(8,20,9), (1,10,21)(2,22,11)(3,12,23)(4,24,13)(5,14,17)(6,18,15)(7,16,19)(8,20,9), (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)(9,18)(10,21)(11,24)(12,19)(13,22)(14,17)(15,20)(16,23) );
 
G=PermutationGroup([[(1,21,10),(2,11,22),(3,12,23),(4,24,13),(5,17,14),(6,15,18),(7,16,19),(8,20,9)], [(1,10,21),(2,11,22),(3,23,12),(4,24,13),(5,14,17),(6,15,18),(7,19,16),(8,20,9)], [(1,10,21),(2,22,11),(3,12,23),(4,24,13),(5,14,17),(6,18,15),(7,16,19),(8,20,9)], [(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),(9,18),(10,21),(11,24),(12,19),(13,22),(14,17),(15,20),(16,23)]])
 
G:=TransitiveGroup(24,1315);
 

Matrix representation of C33⋊6SD16 ►in GL8(𝔽73)

10000000
01000000
00100000
00010000
0000720720
00000100
00001000
00000001
,
10000000
01000000
00100000
00010000
00001000
0000072072
00000010
00000100
,
10000000
01000000
0072720000
00100000
00001000
00000100
00000010
00000001
,
1261000000
60000000
0013430000
0030600000
00000100
00001000
00000001
0000720720
,
10000000
172000000
00010000
00100000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,6,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,13,30,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0],[1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C33⋊6SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_6{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:6SD16");
 
// GroupNames label
 
G:=SmallGroup(432,583);
 
// by ID
 
G=gap.SmallGroup(432,583);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,254,135,58,1684,571,298,677,1027,14118]);
 
// 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=e*a*e=b,b*c=c*b,d*b*d^-1=a^-1,e*b*e=a,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
 
// generators/relations
 

Export

Character table of C33⋊6SD16 in TeX

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