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G = C33⋊5Q8  order 216 = 23·33

3rd semidirect product of C33 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C33⋊5Q8, C32⋊6Dic6, C6.24S32, (C3×C6).37D6, C3⋊Dic3.4S3, C3⋊2(C32⋊2Q8), C2.3(C32⋊4D6), (C32×C6).15C22, (C3×C3⋊Dic3).3C2, SmallGroup(216,133)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C33⋊5Q8
 G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, dad-1=eae-1=a-1, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 252 in 70 conjugacy classes, 23 normal (5 characteristic)
C1, C2, C3, C3, C4, C6, C6, Q8, C32, C32, Dic3, C12, C3×C6, C3×C6, Dic6, C33, C3×Dic3, C3⋊Dic3, C32×C6, C32⋊2Q8, C3×C3⋊Dic3, C33⋊5Q8
Quotients: C1, C2, C22, S3, Q8, D6, Dic6, S32, C32⋊2Q8, C32⋊4D6, C33⋊5Q8

Character table of C33⋊5Q8

 class 123A3B3C3D3E3F3G3H4A4B4C6A6B6C6D6E6F6G6H12A12B12C12D12E12F
 size 112224444418181822244444181818181818
ρ1111111111111111111111111111    trivial
ρ211111111111-1-111111111-1-1-1-111    linear of order 2
ρ31111111111-1-1111111111-111-1-1-1    linear of order 2
ρ41111111111-11-1111111111-1-11-1-1    linear of order 2
ρ5222-12-1-1-1-12-200-122-1-12-1-1000011    orthogonal lifted from D6
ρ62222-1-12-1-1-10202-12-1-1-1-12-100-100    orthogonal lifted from S3
ρ7222-12-1-1-1-12200-122-1-12-1-10000-1-1    orthogonal lifted from S3
ρ822-1222-1-1-1-100222-1-1-1-12-10-1-1000    orthogonal lifted from S3
ρ92222-1-12-1-1-10-202-12-1-1-1-12100100    orthogonal lifted from D6
ρ1022-1222-1-1-1-100-222-1-1-1-12-1011000    orthogonal lifted from D6
ρ112-222222222000-2-2-2-2-2-2-2-2000000    symplectic lifted from Q8, Schur index 2
ρ122-22-12-1-1-1-120001-2-211-2110000√3-√3    symplectic lifted from Dic6, Schur index 2
ρ132-22-12-1-1-1-120001-2-211-2110000-√3√3    symplectic lifted from Dic6, Schur index 2
ρ142-222-1-12-1-1-1000-21-21111-2√300-√300    symplectic lifted from Dic6, Schur index 2
ρ152-2-1222-1-1-1-1000-2-21111-210-√3√3000    symplectic lifted from Dic6, Schur index 2
ρ162-2-1222-1-1-1-1000-2-21111-210√3-√3000    symplectic lifted from Dic6, Schur index 2
ρ172-222-1-12-1-1-1000-21-21111-2-√300√300    symplectic lifted from Dic6, Schur index 2
ρ1844-2-24-2111-2000-24-211-2-21000000    orthogonal lifted from S32
ρ19444-2-21-211-2000-2-2411-21-2000000    orthogonal lifted from S32
ρ2044-24-2-2-21110004-2-2111-2-2000000    orthogonal lifted from S32
ρ214-4-24-2-2-2111000-422-1-1-122000000    symplectic lifted from C32⋊2Q8, Schur index 2
ρ224-44-2-21-211-200022-4-1-12-12000000    symplectic lifted from C32⋊2Q8, Schur index 2
ρ234-4-2-24-2111-20002-42-1-122-1000000    symplectic lifted from C32⋊2Q8, Schur index 2
ρ244-4-2-2-211-1+3√-3/2-1-3√-3/210002221-3√-3/21+3√-3/2-1-1-1000000    complex faithful
ρ2544-2-2-211-1+3√-3/2-1-3√-3/21000-2-2-2-1+3√-3/2-1-3√-3/2111000000    complex lifted from C32⋊4D6
ρ2644-2-2-211-1-3√-3/2-1+3√-3/21000-2-2-2-1-3√-3/2-1+3√-3/2111000000    complex lifted from C32⋊4D6
ρ274-4-2-2-211-1-3√-3/2-1+3√-3/210002221+3√-3/21-3√-3/2-1-1-1000000    complex faithful

Permutation representations of C33⋊5Q8
►On 24 points - transitive group 24T551
Generators in S24
(1 19 14)(2 15 20)(3 17 16)(4 13 18)(5 10 21)(6 22 11)(7 12 23)(8 24 9)

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

(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 21)(6 11 22)(7 12 23)(8 9 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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
 
G:=sub<Sym(24)| (1,19,14)(2,15,20)(3,17,16)(4,13,18)(5,10,21)(6,22,11)(7,12,23)(8,24,9), (1,19,14)(2,15,20)(3,17,16)(4,13,18)(5,21,10)(6,11,22)(7,23,12)(8,9,24), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;
 
G:=Group( (1,19,14)(2,15,20)(3,17,16)(4,13,18)(5,10,21)(6,22,11)(7,12,23)(8,24,9), (1,19,14)(2,15,20)(3,17,16)(4,13,18)(5,21,10)(6,11,22)(7,23,12)(8,9,24), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );
 
G=PermutationGroup([[(1,19,14),(2,15,20),(3,17,16),(4,13,18),(5,10,21),(6,22,11),(7,12,23),(8,24,9)], [(1,19,14),(2,15,20),(3,17,16),(4,13,18),(5,21,10),(6,11,22),(7,23,12),(8,9,24)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)]])
 
G:=TransitiveGroup(24,551);
 

C33⋊5Q8 is a maximal subgroup of
 C33⋊7SD16  C33⋊Q16  C33⋊8SD16  C33⋊3Q16  S3×C32⋊2Q8  C33⋊6(C2×Q8)  (S3×C6).D6  D6⋊S3⋊S3  C3⋊S3⋊4Dic6  C12.95S32  C62.96D6
C33⋊5Q8 is a maximal quotient of
 C62.85D6

Matrix representation of C33⋊5Q8 ►in GL4(𝔽7) generated by

5323
1330
4406
0004
,
3632
6342
0020
0004
,
0526
0202
3361
0004
,
4066
4565
5216
5524
,
3613
5133
3331
2560
G:=sub<GL(4,GF(7))| [5,1,4,0,3,3,4,0,2,3,0,0,3,0,6,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[4,4,5,5,0,5,2,5,6,6,1,2,6,5,6,4],[3,5,3,2,6,1,3,5,1,3,3,6,3,3,1,0] >;
 

C33⋊5Q8 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_5Q_8
 
% in TeX
 
G:=Group("C3^3:5Q8");
 
// GroupNames label
 
G:=SmallGroup(216,133);
 
// by ID
 
G=gap.SmallGroup(216,133);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,387,201,730,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Character table of C33⋊5Q8 in TeX

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