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G = C33⋊9D4  order 216 = 23·33

6th semidirect product of C33 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C33⋊9D4, C32⋊6D12, C6.23S32, C3⋊Dic3⋊6S3, (C3×C6).36D6, C3⋊1(D6⋊S3), C3⋊3(C3⋊D12), C32⋊8(C3⋊D4), C2.2(C32⋊4D6), (C32×C6).14C22, (C2×C3⋊S3)⋊6S3, (C6×C3⋊S3)⋊4C2, (C3×C3⋊Dic3)⋊3C2, SmallGroup(216,132)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊9D4
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9D4
Lower central
C33 — C32×C6 — C33⋊9D4
Upper central
C1 — C2

Generators and relations for C33⋊9D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 428 in 94 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C3⋊D4, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C32×C6, D6⋊S3, C3⋊D12, C3×C3⋊Dic3, C6×C3⋊S3, C33⋊9D4
Quotients: C1, C2, C22, S3, D4, D6, D12, C3⋊D4, S32, D6⋊S3, C3⋊D12, C32⋊4D6, C33⋊9D4

Character table of C33⋊9D4

 class 12A2B2C3A3B3C3D3E3F3G3H46A6B6C6D6E6F6G6H6I6J6K6L12A12B
 size 111818222444441822244444181818181818
ρ1111111111111111111111111111    trivial
ρ211-1111111111-1111111111-1-11-1-1    linear of order 2
ρ3111-111111111-111111111-111-1-1-1    linear of order 2
ρ411-1-111111111111111111-1-1-1-111    linear of order 2
ρ5220-2-1222-1-1-1-102-12-1-1-1-12100100    orthogonal lifted from D6
ρ622202-12-1-12-1-1022-1-1-1-12-10-1-1000    orthogonal lifted from S3
ρ7220022-1-12-1-1-1-2-122-1-12-1-1000011    orthogonal lifted from D6
ρ822-202-12-1-12-1-1022-1-1-1-12-1011000    orthogonal lifted from D6
ρ92202-1222-1-1-1-102-12-1-1-1-12-100-100    orthogonal lifted from S3
ρ10220022-1-12-1-1-12-122-1-12-1-10000-1-1    orthogonal lifted from S3
ρ112-200222222220-2-2-2-2-2-2-2-2000000    orthogonal lifted from D4
ρ122-20022-1-12-1-1-101-2-211-2110000-√3√3    orthogonal lifted from D12
ρ132-20022-1-12-1-1-101-2-211-2110000√3-√3    orthogonal lifted from D12
ρ142-2002-12-1-12-1-10-2-21111-210√-3-√-3000    complex lifted from C3⋊D4
ρ152-200-1222-1-1-1-10-21-21111-2-√-300√-300    complex lifted from C3⋊D4
ρ162-200-1222-1-1-1-10-21-21111-2√-300-√-300    complex lifted from C3⋊D4
ρ172-2002-12-1-12-1-10-2-21111-210-√-3√-3000    complex lifted from C3⋊D4
ρ184400-2-24-21-21104-2-2111-2-2000000    orthogonal lifted from S32
ρ1944004-2-21-2-2110-24-211-2-21000000    orthogonal lifted from S32
ρ204400-24-2-2-21110-2-2411-21-2000000    orthogonal lifted from S32
ρ214-400-24-2-2-2111022-4-1-12-12000000    orthogonal lifted from C3⋊D12
ρ224-4004-2-21-2-21102-42-1-122-1000000    orthogonal lifted from C3⋊D12
ρ234-400-2-24-21-2110-422-1-1-122000000    symplectic lifted from D6⋊S3, Schur index 2
ρ244-400-2-2-2111-1+3√-3/2-1-3√-3/202221-3√-3/21+3√-3/2-1-1-1000000    complex faithful
ρ254400-2-2-2111-1+3√-3/2-1-3√-3/20-2-2-2-1+3√-3/2-1-3√-3/2111000000    complex lifted from C32⋊4D6
ρ264-400-2-2-2111-1-3√-3/2-1+3√-3/202221+3√-3/21-3√-3/2-1-1-1000000    complex faithful
ρ274400-2-2-2111-1-3√-3/2-1+3√-3/20-2-2-2-1-3√-3/2-1+3√-3/2111000000    complex lifted from C32⋊4D6

Permutation representations of C33⋊9D4
►On 24 points - transitive group 24T550
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 14 10)(6 15 11)(7 16 12)(8 13 9)

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

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

(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)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
 
G:=sub<Sym(24)| (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,14,10)(6,15,11)(7,16,12)(8,13,9), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (1,21,20)(2,17,22)(3,23,18)(4,19,24)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (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)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;
 
G:=Group( (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,14,10)(6,15,11)(7,16,12)(8,13,9), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (1,21,20)(2,17,22)(3,23,18)(4,19,24)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (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)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );
 
G=PermutationGroup([[(1,20,21),(2,17,22),(3,18,23),(4,19,24),(5,14,10),(6,15,11),(7,16,12),(8,13,9)], [(1,20,21),(2,22,17),(3,18,23),(4,24,19),(5,14,10),(6,11,15),(7,16,12),(8,9,13)], [(1,21,20),(2,17,22),(3,23,18),(4,19,24),(5,14,10),(6,11,15),(7,16,12),(8,9,13)], [(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),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)]])
 
G:=TransitiveGroup(24,550);
 

C33⋊9D4 is a maximal subgroup of
 C33⋊D8  C33⋊6SD16  C32⋊2D24  C33⋊8SD16  S3×D6⋊S3  S3×C3⋊D12  (S3×C6)⋊D6  D6.S32  D6.6S32  Dic3.S32  C12⋊S3⋊12S3  C12.95S32  C12⋊3S32  C62.96D6  C62⋊24D6
C33⋊9D4 is a maximal quotient of
 C33⋊9D8  C33⋊18SD16  C33⋊9Q16  C62.84D6  C62.85D6

Matrix representation of C33⋊9D4 ►in GL4(𝔽7) generated by

5323
1330
4406
0004
,
3632
6342
0020
0004
,
0526
0202
3361
0004
,
6505
0043
1635
1135
,
2634
0465
4416
5210
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],[6,0,1,1,5,0,6,1,0,4,3,3,5,3,5,5],[2,0,4,5,6,4,4,2,3,6,1,1,4,5,6,0] >;
 

C33⋊9D4 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_9D_4
 
% in TeX
 
G:=Group("C3^3:9D4");
 
// GroupNames label
 
G:=SmallGroup(216,132);
 
// by ID
 
G=gap.SmallGroup(216,132);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,387,201,730,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C33⋊9D4 in TeX

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