Copied to
clipboard

G = D12.S3order 144 = 24·32

1st non-split extension by D12 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D12.1S3, C12.12D6, C6.13D12, C324SD16, C3⋊C82S3, C4.2S32, (C3×C6).9D4, C33(C24⋊C2), C31(D4.S3), (C3×D12).2C2, C6.2(C3⋊D4), C324Q82C2, (C3×C12).4C22, C2.5(C3⋊D12), (C3×C3⋊C8)⋊2C2, SmallGroup(144,59)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.S3
C1C3C32C3×C6C3×C12C3×D12 — D12.S3
C32C3×C6C3×C12 — D12.S3
C1C2C4

Generators and relations for D12.S3
 G = < a,b,c,d | a12=b2=c3=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

12C2
2C3
6C22
18C4
2C6
4S3
12C6
3C8
3D4
9Q8
2C12
2D6
6Dic3
6Dic3
6Dic3
6Dic3
6C2×C6
4C3×S3
9SD16
3C24
3Dic6
3Dic6
3C3×D4
6Dic6
2C3⋊Dic3
2S3×C6
3D4.S3
3C24⋊C2

Character table of D12.S3

 class 12A2B3A3B3C4A4B6A6B6C6D6E8A8B12A12B12C12D12E24A24B24C24D
 size 1112224236224121266224446666
ρ1111111111111111111111111    trivial
ρ211-111111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ31111111-111111-1-111111-1-1-1-1    linear of order 2
ρ411-11111-1111-1-111111111111    linear of order 2
ρ522-22-1-1202-1-1110022-1-1-10000    orthogonal lifted from D6
ρ62222-1-1202-1-1-1-10022-1-1-10000    orthogonal lifted from S3
ρ7220222-202220000-2-2-2-2-20000    orthogonal lifted from D4
ρ8220-12-120-12-100-2-2-1-1-1-121111    orthogonal lifted from D6
ρ9220-12-120-12-10022-1-1-1-12-1-1-1-1    orthogonal lifted from S3
ρ10220-12-1-20-12-100001111-2-3-333    orthogonal lifted from D12
ρ11220-12-1-20-12-100001111-233-3-3    orthogonal lifted from D12
ρ122-2022200-2-2-200-2--200000-2--2--2-2    complex lifted from SD16
ρ132-2022200-2-2-200--2-200000--2-2-2--2    complex lifted from SD16
ρ142202-1-1-202-1-1-3--300-2-21110000    complex lifted from C3⋊D4
ρ152202-1-1-202-1-1--3-300-2-21110000    complex lifted from C3⋊D4
ρ162-20-12-1001-2100--2-23-33-30ζ83ζ3838ζ3ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32    complex lifted from C24⋊C2
ρ172-20-12-1001-2100-2--23-33-30ζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32    complex lifted from C24⋊C2
ρ182-20-12-1001-2100--2-2-33-330ζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3    complex lifted from C24⋊C2
ρ192-20-12-1001-2100-2--2-33-330ζ87ζ328785ζ32ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3    complex lifted from C24⋊C2
ρ20440-2-21-40-2-21000022-1-120000    orthogonal lifted from C3⋊D12
ρ21440-2-2140-2-210000-2-211-20000    orthogonal lifted from S32
ρ224-404-2-200-4220000000000000    symplectic lifted from D4.S3, Schur index 2
ρ234-40-2-210022-1000023-23-3300000    symplectic faithful, Schur index 2
ρ244-40-2-210022-10000-23233-300000    symplectic faithful, Schur index 2

Smallest permutation representation of D12.S3
On 48 points
Generators in S48
(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)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 39)(14 38)(15 37)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 13 10 22 7 19 4 16)(2 14 11 23 8 20 5 17)(3 15 12 24 9 21 6 18)(25 37 34 46 31 43 28 40)(26 38 35 47 32 44 29 41)(27 39 36 48 33 45 30 42)

G:=sub<Sym(48)| (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,39)(14,38)(15,37)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,13,10,22,7,19,4,16)(2,14,11,23,8,20,5,17)(3,15,12,24,9,21,6,18)(25,37,34,46,31,43,28,40)(26,38,35,47,32,44,29,41)(27,39,36,48,33,45,30,42)>;

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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,39)(14,38)(15,37)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,13,10,22,7,19,4,16)(2,14,11,23,8,20,5,17)(3,15,12,24,9,21,6,18)(25,37,34,46,31,43,28,40)(26,38,35,47,32,44,29,41)(27,39,36,48,33,45,30,42) );

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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,39),(14,38),(15,37),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,13,10,22,7,19,4,16),(2,14,11,23,8,20,5,17),(3,15,12,24,9,21,6,18),(25,37,34,46,31,43,28,40),(26,38,35,47,32,44,29,41),(27,39,36,48,33,45,30,42)])

D12.S3 is a maximal subgroup of
S3×C24⋊C2  D24⋊S3  C24.3D6  D247S3  D12.27D6  D12.28D6  D12.29D6  D12.D6  S3×D4.S3  D129D6  D12.8D6  D12.9D6  D12.11D6  D12.12D6  D12.15D6  D36.S3  C36.D6  He33SD16  He34SD16  C3314SD16  C3316SD16  C3318SD16
D12.S3 is a maximal quotient of
C6.16D24  C12.73D12  C12.Dic6  D36.S3  C36.D6  He35SD16  C3314SD16  C3316SD16  C3318SD16

Matrix representation of D12.S3 in GL6(𝔽73)

4600000
27270000
0072100
0072000
000010
000001
,
46190000
27270000
0072000
0072100
000010
000001
,
100000
010000
001000
000100
0000721
0000720
,
2200000
57630000
0072000
0007200
000001
000010

G:=sub<GL(6,GF(73))| [46,27,0,0,0,0,0,27,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,27,0,0,0,0,19,27,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[22,57,0,0,0,0,0,63,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D12.S3 in GAP, Magma, Sage, TeX

D_{12}.S_3
% in TeX

G:=Group("D12.S3");
// GroupNames label

G:=SmallGroup(144,59);
// by ID

G=gap.SmallGroup(144,59);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,31,218,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^3=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D12.S3 in TeX
Character table of D12.S3 in TeX

׿
×
𝔽