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G = C322D12order 216 = 23·33

The semidirect product of C32 and D12 acting via D12/C3=D4

non-abelian, soluble, monomial

Aliases: C333D4, C322D12, C32⋊C4⋊S3, C31S3≀C2, C3⋊S3.2D6, C324D62C2, (C3×C32⋊C4)⋊1C2, (C3×C3⋊S3).5C22, SmallGroup(216,159)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊S3 — C322D12
C1C3C33C3×C3⋊S3C324D6 — C322D12
C33C3×C3⋊S3 — C322D12
C1

Generators and relations for C322D12
 G = < a,b,c,d | a3=b3=c12=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

Subgroups: 436 in 60 conjugacy classes, 11 normal (9 characteristic)
C1, C2 [×3], C3, C3 [×4], C4, C22 [×2], S3 [×8], C6 [×3], D4, C32, C32 [×4], C12, D6 [×4], C3×S3 [×8], C3⋊S3, C3⋊S3 [×2], D12, C33, C32⋊C4, S32 [×4], C3×C3⋊S3, C3×C3⋊S3 [×2], S3≀C2, C3×C32⋊C4, C324D6 [×2], C322D12
Quotients: C1, C2 [×3], C22, S3, D4, D6, D12, S3≀C2, C322D12

Character table of C322D12

 class 12A2B2C3A3B3C3D3E46A6B6C12A12B
 size 19181824488181836361818
ρ1111111111111111    trivial
ρ211-1-11111111-1-111    linear of order 2
ρ311-1111111-111-1-1-1    linear of order 2
ρ4111-111111-11-11-1-1    linear of order 2
ρ52200-122-1-12-100-1-1    orthogonal lifted from S3
ρ62200-122-1-1-2-10011    orthogonal lifted from D6
ρ72-200222220-20000    orthogonal lifted from D4
ρ82-200-122-1-10100-33    orthogonal lifted from D12
ρ92-200-122-1-101003-3    orthogonal lifted from D12
ρ1040-204-21-21000100    orthogonal lifted from S3≀C2
ρ11400241-21-200-1000    orthogonal lifted from S3≀C2
ρ1240204-21-21000-100    orthogonal lifted from S3≀C2
ρ13400-241-21-2001000    orthogonal lifted from S3≀C2
ρ148000-4-422-1000000    orthogonal faithful
ρ158000-42-4-12000000    orthogonal faithful

Permutation representations of C322D12
On 12 points - transitive group 12T118
Generators in S12
(2 6 10)(4 12 8)
(1 5 9)(3 11 7)
(1 2 3 4 5 6 7 8 9 10 11 12)
(1 3)(4 12)(5 11)(6 10)(7 9)

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

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

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

G:=TransitiveGroup(12,118);

On 18 points - transitive group 18T104
Generators in S18
(1 14 8)(2 15 9)(3 10 16)(4 11 17)(5 18 12)(6 7 13)
(1 14 8)(2 9 15)(3 10 16)(4 17 11)(5 18 12)(6 13 7)
(1 2 3 4 5 6)(7 8 9 10 11 12 13 14 15 16 17 18)
(1 2)(3 6)(4 5)(7 16)(8 15)(9 14)(10 13)(11 12)(17 18)

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

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

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

G:=TransitiveGroup(18,104);

On 24 points - transitive group 24T558
Generators in S24
(1 5 9)(3 11 7)(14 18 22)(16 24 20)
(2 10 6)(4 8 12)(13 17 21)(15 23 19)
(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 14)(2 13)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)

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

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

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

G:=TransitiveGroup(24,558);

On 27 points - transitive group 27T79
Generators in S27
(1 13 7)(2 5 11)(3 9 15)(4 22 25)(6 27 24)(8 26 17)(10 19 16)(12 18 21)(14 23 20)
(1 4 10)(2 8 14)(3 12 6)(5 26 23)(7 25 16)(9 18 27)(11 17 20)(13 22 19)(15 21 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 25 26 27)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)

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

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

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

G:=TransitiveGroup(27,79);

C322D12 is a maximal subgroup of   C333SD16  F9⋊S3  S3×S3≀C2
C322D12 is a maximal quotient of   (C3×C6).8D12  (C3×C6).9D12  C322D24  C338SD16  C333Q16

Polynomial with Galois group C322D12 over ℚ
actionf(x)Disc(f)
12T118x12-2x9-2x6+3x3-1-315·136

Matrix representation of C322D12 in GL6(𝔽13)

100000
010000
0000012
000100
000010
0010012
,
100000
010000
001000
0000120
0001120
000001
,
7100000
3100000
0001200
0000012
0012000
0000120
,
7100000
360000
0000012
0001200
0000120
0012000

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1],[7,3,0,0,0,0,10,10,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0],[7,3,0,0,0,0,10,6,0,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0,0,0] >;

C322D12 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2D_{12}
% in TeX

G:=Group("C3^2:2D12");
// GroupNames label

G:=SmallGroup(216,159);
// by ID

G=gap.SmallGroup(216,159);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,73,31,579,585,111,244,130,376,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^12=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of C322D12 in TeX

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