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G = C3⋊D12order 72 = 23·32

The semidirect product of C3 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D62S3, C32D12, Dic3⋊S3, C6.4D6, C323D4, C2.4S32, (S3×C6)⋊2C2, C31(C3⋊D4), (C3×Dic3)⋊1C2, (C3×C6).4C22, (C2×C3⋊S3)⋊1C2, SmallGroup(72,23)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3⋊D12
C1C3C32C3×C6S3×C6 — C3⋊D12
C32C3×C6 — C3⋊D12
C1C2

Generators and relations for C3⋊D12
 G = < a,b,c | a3=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >

6C2
18C2
2C3
3C4
3C22
9C22
2C6
2S3
6S3
6C6
6S3
6S3
6S3
9D4
3C2×C6
3D6
3D6
3C12
6D6
2C3×S3
2C3⋊S3
3D12
3C3⋊D4

Character table of C3⋊D12

 class 12A2B2C3A3B3C46A6B6C6D6E12A12B
 size 1161822462246666
ρ1111111111111111    trivial
ρ211-1-11111111-1-111    linear of order 2
ρ311-11111-1111-1-1-1-1    linear of order 2
ρ4111-1111-111111-1-1    linear of order 2
ρ522202-1-10-12-1-1-100    orthogonal lifted from S3
ρ622-202-1-10-12-11100    orthogonal lifted from D6
ρ72-2002220-2-2-20000    orthogonal lifted from D4
ρ82200-12-122-1-100-1-1    orthogonal lifted from S3
ρ92200-12-1-22-1-10011    orthogonal lifted from D6
ρ102-200-12-10-211003-3    orthogonal lifted from D12
ρ112-200-12-10-21100-33    orthogonal lifted from D12
ρ122-2002-1-101-21--3-300    complex lifted from C3⋊D4
ρ132-2002-1-101-21-3--300    complex lifted from C3⋊D4
ρ144400-2-210-2-210000    orthogonal lifted from S32
ρ154-400-2-21022-10000    orthogonal faithful

Permutation representations of C3⋊D12
On 12 points - transitive group 12T38
Generators in S12
(1 9 5)(2 6 10)(3 11 7)(4 8 12)
(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)| (1,9,5)(2,6,10)(3,11,7)(4,8,12), (1,2,3,4,5,6,7,8,9,10,11,12), (1,3)(4,12)(5,11)(6,10)(7,9)>;

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

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

G:=TransitiveGroup(12,38);

On 24 points - transitive group 24T74
Generators in S24
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 21 17)(14 18 22)(15 23 19)(16 20 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 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)(2,10,6)(3,7,11)(4,12,8)(13,21,17)(14,18,22)(15,23,19)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,14)(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)(2,10,6)(3,7,11)(4,12,8)(13,21,17)(14,18,22)(15,23,19)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,14)(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),(2,10,6),(3,7,11),(4,12,8),(13,21,17),(14,18,22),(15,23,19),(16,20,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,14),(2,13),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15)])

G:=TransitiveGroup(24,74);

C3⋊D12 is a maximal subgroup of
D12⋊S3  D6.D6  D6.6D6  S3×D12  D6.3D6  S3×C3⋊D4  Dic3⋊D6  C3⋊D36  C9⋊D12  He32D4  He33D4  C337D4  C338D4  C339D4  GL2(𝔽3)⋊S3  D6.2S4  Dic3⋊S4  A4⋊D12  C3⋊D60  D62D15  D30⋊S3
C3⋊D12 is a maximal quotient of
C3⋊D24  D12.S3  C325SD16  C323Q16  D6⋊Dic3  C6.D12  Dic3⋊Dic3  C3⋊D36  C9⋊D12  He33D4  C337D4  C338D4  C339D4  Dic3⋊S4  A4⋊D12  C3⋊D60  D62D15  D30⋊S3

Polynomial with Galois group C3⋊D12 over ℚ
actionf(x)Disc(f)
12T38x12-36x10+486x8-3084x6+9585x4-13608x2+6534235·339·116·296

Matrix representation of C3⋊D12 in GL4(ℤ) generated by

0-100
1-100
00-11
00-10
,
000-1
001-1
0100
-1100
,
-1100
0100
001-1
000-1
G:=sub<GL(4,Integers())| [0,1,0,0,-1,-1,0,0,0,0,-1,-1,0,0,1,0],[0,0,0,-1,0,0,1,1,0,1,0,0,-1,-1,0,0],[-1,0,0,0,1,1,0,0,0,0,1,0,0,0,-1,-1] >;

C3⋊D12 in GAP, Magma, Sage, TeX

C_3\rtimes D_{12}
% in TeX

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

G:=SmallGroup(72,23);
// by ID

G=gap.SmallGroup(72,23);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,61,26,168,1204]);
// Polycyclic

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

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Subgroup lattice of C3⋊D12 in TeX
Character table of C3⋊D12 in TeX

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