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G = C12⋊S3order 72 = 23·32

1st semidirect product of C12 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C121S3, C31D12, C325D4, C6.14D6, C4⋊(C3⋊S3), (C3×C12)⋊1C2, (C3×C6).13C22, (C2×C3⋊S3)⋊2C2, C2.4(C2×C3⋊S3), SmallGroup(72,33)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12⋊S3
C1C3C32C3×C6C2×C3⋊S3 — C12⋊S3
C32C3×C6 — C12⋊S3
C1C2C4

Generators and relations for C12⋊S3
 G = < a,b,c | a12=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

18C2
18C2
9C22
9C22
6S3
6S3
6S3
6S3
6S3
6S3
6S3
6S3
9D4
3D6
3D6
3D6
3D6
3D6
3D6
3D6
3D6
2C3⋊S3
2C3⋊S3
3D12
3D12
3D12
3D12

Character table of C12⋊S3

 class 12A2B2C3A3B3C3D46A6B6C6D12A12B12C12D12E12F12G12H
 size 11181822222222222222222
ρ1111111111111111111111    trivial
ρ211-1-111111111111111111    linear of order 2
ρ311-111111-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-11111-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ52200-12-1-12-1-12-1-1-1-1-1-1-122    orthogonal lifted from S3
ρ62200-1-12-1-2-1-1-12-211-21111    orthogonal lifted from D6
ρ72200-12-1-1-2-1-12-1111111-2-2    orthogonal lifted from D6
ρ822002-1-1-1-2-12-1-111-21-2111    orthogonal lifted from D6
ρ922002-1-1-12-12-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ102200-1-12-12-1-1-122-1-12-1-1-1-1    orthogonal lifted from S3
ρ112200-1-1-12-22-1-1-11-2111-211    orthogonal lifted from D6
ρ122-20022220-2-2-2-200000000    orthogonal lifted from D4
ρ132200-1-1-1222-1-1-1-12-1-1-12-1-1    orthogonal lifted from S3
ρ142-200-1-1-120-2111-3033-303-3    orthogonal lifted from D12
ρ152-200-12-1-1011-21333-3-3-300    orthogonal lifted from D12
ρ162-2002-1-1-101-211-33030-3-33    orthogonal lifted from D12
ρ172-200-12-1-1011-21-3-3-333300    orthogonal lifted from D12
ρ182-200-1-12-10111-203-303-33-3    orthogonal lifted from D12
ρ192-200-1-12-10111-20-330-33-33    orthogonal lifted from D12
ρ202-2002-1-1-101-2113-30-3033-3    orthogonal lifted from D12
ρ212-200-1-1-120-211130-3-330-33    orthogonal lifted from D12

Smallest permutation representation of C12⋊S3
On 36 points
Generators in S36
(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 28 29 30 31 32 33 34 35 36)
(1 16 31)(2 17 32)(3 18 33)(4 19 34)(5 20 35)(6 21 36)(7 22 25)(8 23 26)(9 24 27)(10 13 28)(11 14 29)(12 15 30)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)

G:=sub<Sym(36)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,16,31)(2,17,32)(3,18,33)(4,19,34)(5,20,35)(6,21,36)(7,22,25)(8,23,26)(9,24,27)(10,13,28)(11,14,29)(12,15,30), (2,12)(3,11)(4,10)(5,9)(6,8)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,16,31)(2,17,32)(3,18,33)(4,19,34)(5,20,35)(6,21,36)(7,22,25)(8,23,26)(9,24,27)(10,13,28)(11,14,29)(12,15,30), (2,12)(3,11)(4,10)(5,9)(6,8)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)], [(1,16,31),(2,17,32),(3,18,33),(4,19,34),(5,20,35),(6,21,36),(7,22,25),(8,23,26),(9,24,27),(10,13,28),(11,14,29),(12,15,30)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,36),(24,35)])

C12⋊S3 is a maximal subgroup of
C3⋊D24  C325SD16  C242S3  C325D8  C327D8  C3211SD16  D6.6D6  S3×D12  C12.59D6  D4×C3⋊S3  C12.26D6  He34D4  C36⋊S3  C338D4  C3312D4  C12⋊S4  C12.7S4  C15⋊D12  C60⋊S3
C12⋊S3 is a maximal quotient of
C242S3  C325D8  C325Q16  C12⋊Dic3  C6.11D12  C36⋊S3  He35D4  C338D4  C3312D4  C12⋊S4  C15⋊D12  C60⋊S3

Matrix representation of C12⋊S3 in GL4(𝔽13) generated by

0100
121200
0063
00103
,
1000
0100
001212
0010
,
1000
121200
001212
0001
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,6,10,0,0,3,3],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,12,0],[1,12,0,0,0,12,0,0,0,0,12,0,0,0,12,1] >;

C12⋊S3 in GAP, Magma, Sage, TeX

C_{12}\rtimes S_3
% in TeX

G:=Group("C12:S3");
// GroupNames label

G:=SmallGroup(72,33);
// by ID

G=gap.SmallGroup(72,33);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,61,26,323,1204]);
// Polycyclic

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

Export

Subgroup lattice of C12⋊S3 in TeX
Character table of C12⋊S3 in TeX

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