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G = D12.S3  order 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, C32⋊4SD16, C3⋊C8⋊2S3, C4.2S32, (C3×C6).9D4, C3⋊3(C24⋊C2), C3⋊1(D4.S3), (C3×D12).2C2, C6.2(C3⋊D4), C32⋊4Q8⋊2C2, (C3×C12).4C22, C2.5(C3⋊D12), (C3×C3⋊C8)⋊2C2, SmallGroup(144,59)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.S3
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12.S3
Lower central
C32 — C3×C6 — C3×C12 — D12.S3
Upper central
C1 — C2 — C4

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 >

C1
C2
12C2
C3
C3
2C3
C4
6C22
18C4
C6
C6
2C6
4S3
12C6
C32
3C8
3D4
9Q8
C12
C12
2C12
2D6
6Dic3
6Dic3
6Dic3
6Dic3
6C2×C6
C3×C6
4C3×S3
9SD16
D12
C3⋊C8
3C24
3Dic6
3Dic6
3C3×D4
6Dic6
C3×C12
2C3⋊Dic3
2S3×C6
3D4.S3
3C24⋊C2
C3×C3⋊C8
C3×D12
C32⋊4Q8
D12.S3

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-√3√3√3    orthogonal lifted from D12
ρ11220-12-1-20-12-100001111-2√3√3-√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√-2√3-√3√3-√30ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32    complex lifted from C24⋊C2
ρ172-20-12-1001-2100√-2-√-2√3-√3√3-√30ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32    complex lifted from C24⋊C2
ρ182-20-12-1001-2100-√-2√-2-√3√3-√3√30ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3    complex lifted from C24⋊C2
ρ192-20-12-1001-2100√-2-√-2-√3√3-√3√30ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3    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
ρ234-40-2-210022-100002√3-2√3-√3√300000    symplectic faithful, Schur index 2
ρ244-40-2-210022-10000-2√32√3√3-√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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)

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

D12.S3 is a maximal subgroup of
 S3×C24⋊C2  D24⋊S3  C24.3D6  D24⋊7S3  D12.27D6  D12.28D6  D12.29D6  D12.D6  S3×D4.S3  D12⋊9D6  D12.8D6  D12.9D6  D12.11D6  D12.12D6  D12.15D6  D36.S3  C36.D6  He3⋊3SD16  He3⋊4SD16  C33⋊14SD16  C33⋊16SD16  C33⋊18SD16
D12.S3 is a maximal quotient of
 C6.16D24  C12.73D12  C12.Dic6  D36.S3  C36.D6  He3⋊5SD16  C33⋊14SD16  C33⋊16SD16  C33⋊18SD16

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

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