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G = C339D4order 216 = 23·33

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

metabelian, supersoluble, monomial

Aliases: C339D4, C326D12, C6.23S32, C3⋊Dic36S3, (C3×C6).36D6, C31(D6⋊S3), C33(C3⋊D12), C328(C3⋊D4), C2.2(C324D6), (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

C1C32×C6 — C339D4
C1C3C32C33C32×C6C6×C3⋊S3 — C339D4
C33C32×C6 — C339D4
C1C2

Generators and relations for C339D4
 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 [×2], C3, C3 [×2], C3 [×4], C4, C22 [×2], S3 [×8], C6, C6 [×2], C6 [×6], D4, C32, C32 [×2], C32 [×4], Dic3 [×3], C12, D6 [×6], C2×C6 [×2], C3×S3 [×8], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], D12, C3⋊D4 [×2], C33, C3×Dic3 [×3], C3⋊Dic3, S3×C6 [×6], C2×C3⋊S3 [×2], C3×C3⋊S3 [×2], C32×C6, D6⋊S3, C3⋊D12 [×2], C3×C3⋊Dic3, C6×C3⋊S3 [×2], C339D4
Quotients: C1, C2 [×3], C22, S3 [×3], D4, D6 [×3], D12, C3⋊D4 [×2], S32 [×3], D6⋊S3, C3⋊D12 [×2], C324D6, C339D4

Character table of C339D4

 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
ρ122-20022-1-12-1-1-101-2-211-2110000-33    orthogonal lifted from D12
ρ132-20022-1-12-1-1-101-2-211-21100003-3    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 C324D6
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 C324D6

Permutation representations of C339D4
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);

C339D4 is a maximal subgroup of
C33⋊D8  C336SD16  C322D24  C338SD16  S3×D6⋊S3  S3×C3⋊D12  (S3×C6)⋊D6  D6.S32  D6.6S32  Dic3.S32  C12⋊S312S3  C12.95S32  C123S32  C62.96D6  C6224D6
C339D4 is a maximal quotient of
C339D8  C3318SD16  C339Q16  C62.84D6  C62.85D6

Matrix representation of C339D4 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] >;

C339D4 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 C339D4 in TeX

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