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G = C324D6order 108 = 22·33

The semidirect product of C32 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, A-group

Aliases: C324D6, C333C22, C32S32, C3⋊S32S3, (C3×C3⋊S3)⋊3C2, SmallGroup(108,40)

Series: Derived Chief Lower central Upper central

C1C33 — C324D6
C1C3C32C33C3×C3⋊S3 — C324D6
C33 — C324D6
C1

Generators and relations for C324D6
 G = < a,b,c,d | a3=b3=c6=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 224 in 54 conjugacy classes, 15 normal (3 characteristic)
C1, C2 [×3], C3 [×3], C3 [×4], C22, S3 [×9], C6 [×3], C32 [×3], C32 [×4], D6 [×3], C3×S3 [×9], C3⋊S3 [×3], C33, S32 [×3], C3×C3⋊S3 [×3], C324D6
Quotients: C1, C2 [×3], C22, S3 [×3], D6 [×3], S32 [×3], C324D6

Character table of C324D6

 class 12A2B2C3A3B3C3D3E3F3G3H6A6B6C
 size 199922244444181818
ρ1111111111111111    trivial
ρ21-1-11111111111-1-1    linear of order 2
ρ31-11-111111111-11-1    linear of order 2
ρ411-1-111111111-1-11    linear of order 2
ρ5220022-1-1-1-1-1200-1    orthogonal lifted from S3
ρ620-20-122-1-1-12-1010    orthogonal lifted from D6
ρ72-20022-1-1-1-1-12001    orthogonal lifted from D6
ρ82020-122-1-1-12-10-10    orthogonal lifted from S3
ρ920022-12-1-12-1-1-100    orthogonal lifted from S3
ρ10200-22-12-1-12-1-1100    orthogonal lifted from D6
ρ1140004-2-211-21-2000    orthogonal lifted from S32
ρ124000-24-2111-2-2000    orthogonal lifted from S32
ρ134000-2-2411-2-21000    orthogonal lifted from S32
ρ144000-2-2-2-1-3-3/2-1+3-3/2111000    complex faithful
ρ154000-2-2-2-1+3-3/2-1-3-3/2111000    complex faithful

Permutation representations of C324D6
On 12 points - transitive group 12T71
Generators in S12
(1 5 3)(2 4 6)(7 11 9)(8 10 12)
(1 5 3)(2 4 6)(7 9 11)(8 12 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 10)(2 9)(3 8)(4 7)(5 12)(6 11)

G:=sub<Sym(12)| (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11)>;

G:=Group( (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11) );

G=PermutationGroup([(1,5,3),(2,4,6),(7,11,9),(8,10,12)], [(1,5,3),(2,4,6),(7,9,11),(8,12,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,10),(2,9),(3,8),(4,7),(5,12),(6,11)])

G:=TransitiveGroup(12,71);

On 18 points - transitive group 18T53
Generators in S18
(1 10 7)(2 8 11)(3 12 9)(4 18 15)(5 16 13)(6 14 17)
(1 7 10)(2 11 8)(3 9 12)(4 18 15)(5 16 13)(6 14 17)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 5)(2 4)(3 6)(7 13)(8 18)(9 17)(10 16)(11 15)(12 14)

G:=sub<Sym(18)| (1,10,7)(2,8,11)(3,12,9)(4,18,15)(5,16,13)(6,14,17), (1,7,10)(2,11,8)(3,9,12)(4,18,15)(5,16,13)(6,14,17), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,5)(2,4)(3,6)(7,13)(8,18)(9,17)(10,16)(11,15)(12,14)>;

G:=Group( (1,10,7)(2,8,11)(3,12,9)(4,18,15)(5,16,13)(6,14,17), (1,7,10)(2,11,8)(3,9,12)(4,18,15)(5,16,13)(6,14,17), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,5)(2,4)(3,6)(7,13)(8,18)(9,17)(10,16)(11,15)(12,14) );

G=PermutationGroup([(1,10,7),(2,8,11),(3,12,9),(4,18,15),(5,16,13),(6,14,17)], [(1,7,10),(2,11,8),(3,9,12),(4,18,15),(5,16,13),(6,14,17)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,5),(2,4),(3,6),(7,13),(8,18),(9,17),(10,16),(11,15),(12,14)])

G:=TransitiveGroup(18,53);

On 27 points - transitive group 27T35
Generators in S27
(1 11 14)(2 15 12)(3 13 10)(4 22 18)(5 19 23)(6 24 20)(7 21 25)(8 26 16)(9 17 27)
(1 7 4)(2 5 8)(3 9 6)(10 27 20)(11 21 22)(12 23 16)(13 17 24)(14 25 18)(15 19 26)
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)(16 17 18 19 20 21)(22 23 24 25 26 27)
(1 3)(4 9)(5 8)(6 7)(10 14)(11 13)(16 23)(17 22)(18 27)(19 26)(20 25)(21 24)

G:=sub<Sym(27)| (1,11,14)(2,15,12)(3,13,10)(4,22,18)(5,19,23)(6,24,20)(7,21,25)(8,26,16)(9,17,27), (1,7,4)(2,5,8)(3,9,6)(10,27,20)(11,21,22)(12,23,16)(13,17,24)(14,25,18)(15,19,26), (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15)(16,17,18,19,20,21)(22,23,24,25,26,27), (1,3)(4,9)(5,8)(6,7)(10,14)(11,13)(16,23)(17,22)(18,27)(19,26)(20,25)(21,24)>;

G:=Group( (1,11,14)(2,15,12)(3,13,10)(4,22,18)(5,19,23)(6,24,20)(7,21,25)(8,26,16)(9,17,27), (1,7,4)(2,5,8)(3,9,6)(10,27,20)(11,21,22)(12,23,16)(13,17,24)(14,25,18)(15,19,26), (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15)(16,17,18,19,20,21)(22,23,24,25,26,27), (1,3)(4,9)(5,8)(6,7)(10,14)(11,13)(16,23)(17,22)(18,27)(19,26)(20,25)(21,24) );

G=PermutationGroup([(1,11,14),(2,15,12),(3,13,10),(4,22,18),(5,19,23),(6,24,20),(7,21,25),(8,26,16),(9,17,27)], [(1,7,4),(2,5,8),(3,9,6),(10,27,20),(11,21,22),(12,23,16),(13,17,24),(14,25,18),(15,19,26)], [(1,2,3),(4,5,6,7,8,9),(10,11,12,13,14,15),(16,17,18,19,20,21),(22,23,24,25,26,27)], [(1,3),(4,9),(5,8),(6,7),(10,14),(11,13),(16,23),(17,22),(18,27),(19,26),(20,25),(21,24)])

G:=TransitiveGroup(27,35);

C324D6 is a maximal subgroup of
C33⋊D4  C322D12  S33  C32⋊D18  He35D6  C325D18  C33⋊A4  C3317D6  C6210D6
C324D6 is a maximal quotient of
C339(C2×C4)  C339D4  C335Q8  C325D18  He36D6  He3.6D6  C3317D6  C6210D6

Polynomial with Galois group C324D6 over ℚ
actionf(x)Disc(f)
12T71x12-5x9+11x6-10x3+4210·318·56

Matrix representation of C324D6 in GL4(𝔽7) generated by

5323
1330
4406
0004
,
3632
6342
0020
0004
,
5221
0611
2562
5524
,
5201
3336
3426
5514
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],[5,0,2,5,2,6,5,5,2,1,6,2,1,1,2,4],[5,3,3,5,2,3,4,5,0,3,2,1,1,6,6,4] >;

C324D6 in GAP, Magma, Sage, TeX

C_3^2\rtimes_4D_6
% in TeX

G:=Group("C3^2:4D6");
// GroupNames label

G:=SmallGroup(108,40);
// by ID

G=gap.SmallGroup(108,40);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,122,67,248,1804]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Character table of C324D6 in TeX

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