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G = D12:S3  order 144 = 24·32

3rd semidirect product of D12 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D12:3S3, D6.2D6, Dic6:3S3, C12.22D6, Dic3.1D6, C4.11S32, (C3xD12):5C2, C3:D12:3C2, (C3xDic6):5C2, (S3xDic3):1C2, C32:2(C4oD4), (C3xC6).3C23, C6.3(C22xS3), C3:1(D4:2S3), C3:2(Q8:3S3), (S3xC6).2C22, (C3xC12).18C22, C3:Dic3.11C22, (C3xDic3).2C22, C2.6(C2xS32), (C4xC3:S3):1C2, (C2xC3:S3).11C22, SmallGroup(144,139)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — D12:S3
Chief series
C1 — C3 — C32 — C3xC6 — S3xC6 — S3xDic3 — D12:S3
Lower central
C32 — C3xC6 — D12:S3
Upper central
C1 — C2 — C4

Generators and relations for D12:S3
 G = < a,b,c,d | a12=b2=c3=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 288 in 88 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2xC4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C3xD4, C3xQ8, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, D4:2S3, Q8:3S3, S3xDic3, C3:D12, C3xDic6, C3xD12, C4xC3:S3, D12:S3
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, S32, D4:2S3, Q8:3S3, C2xS32, D12:S3

Character table of D12:S3

 class 12A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E12A12B12C12D12E12F
 size 11661822426699224121244441212
ρ1111111111111111111111111    trivial
ρ211-111111-11-1-1-1111-11-1-1-1-1-11    linear of order 2
ρ311-11-1111-1-1111111-11-1-1-1-11-1    linear of order 2
ρ41111-11111-1-1-1-1111111111-1-1    linear of order 2
ρ511-1-111111-1-111111-1-11111-1-1    linear of order 2
ρ6111-11111-1-11-1-11111-1-1-1-1-11-1    linear of order 2
ρ7111-1-1111-11-1111111-1-1-1-1-1-11    linear of order 2
ρ811-1-1-1111111-1-1111-1-1111111    linear of order 2
ρ922000-12-1-2-2200-12-1001-211-11    orthogonal lifted from D6
ρ1022000-12-12-2-200-12-100-12-1-111    orthogonal lifted from D6
ρ11222202-1-1200002-1-1-1-12-1-1-100    orthogonal lifted from S3
ρ1222-2-202-1-1200002-1-1112-1-1-100    orthogonal lifted from D6
ρ1322-2202-1-1-200002-1-11-1-211100    orthogonal lifted from D6
ρ1422000-12-122200-12-100-12-1-1-1-1    orthogonal lifted from S3
ρ1522000-12-1-22-200-12-1001-2111-1    orthogonal lifted from D6
ρ16222-202-1-1-200002-1-1-11-211100    orthogonal lifted from D6
ρ172-2000222000-2i2i-2-2-200000000    complex lifted from C4oD4
ρ182-20002220002i-2i-2-2-200000000    complex lifted from C4oD4
ρ1944000-2-21-40000-2-210022-1-100    orthogonal lifted from C2xS32
ρ204-4000-24-2000002-4200000000    orthogonal lifted from Q8:3S3, Schur index 2
ρ2144000-2-2140000-2-2100-2-21100    orthogonal lifted from S32
ρ224-40004-2-200000-42200000000    symplectic lifted from D4:2S3, Schur index 2
ρ234-4000-2-210000022-100003i-3i00    complex faithful
ρ244-4000-2-210000022-10000-3i3i00    complex faithful

Permutation representations of D12:S3
►On 24 points - transitive group 24T227
Generators in S24
(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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)

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

(2 6)(3 11)(5 9)(8 12)(13 19)(14 24)(15 17)(16 22)(18 20)(21 23)
 
G:=sub<Sym(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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (2,6)(3,11)(5,9)(8,12)(13,19)(14,24)(15,17)(16,22)(18,20)(21,23)>;
 
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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (2,6)(3,11)(5,9)(8,12)(13,19)(14,24)(15,17)(16,22)(18,20)(21,23) );
 
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)], [(1,15),(2,14),(3,13),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(2,6),(3,11),(5,9),(8,12),(13,19),(14,24),(15,17),(16,22),(18,20),(21,23)]])
 
G:=TransitiveGroup(24,227);
 

D12:S3 is a maximal subgroup of
 C24:6D6  D12.2D6  D24:5S3  D12.4D6  D12.8D6  D12:5D6  D12.14D6  D12.15D6  D12.33D6  D12:23D6  D12:24D6  S3xD4:2S3  D12:13D6  D12.25D6  S3xQ8:3S3  D18.D6  D12:D9  C12:S3:S3  C12.S32  D6.3S32  D6.6S32  D12:(C3:S3)  C12.39S32  C12:S3:12S3
D12:S3 is a maximal quotient of
 C62.13C23  C62.16C23  C62.18C23  C62.19C23  C62.23C23  C62.28C23  C62.32C23  C62.33C23  C12.30D12  C62.42C23  C62.48C23  C62.51C23  C62.54C23  Dic3:D12  D6:1Dic6  D6.D12  D12:Dic3  C62.77C23  C12:2D12  D18.D6  D12:D9  C12.84S32  D6.3S32  D6.6S32  D12:(C3:S3)  C12.39S32  C12:S3:12S3

Matrix representation of D12:S3 ►in GL4(F5) generated by

0341
1004
2444
3401
,
4412
0131
4241
3401
,
3430
0014
4333
4442
,
1020
0041
0040
0140
G:=sub<GL(4,GF(5))| [0,1,2,3,3,0,4,4,4,0,4,0,1,4,4,1],[4,0,4,3,4,1,2,4,1,3,4,0,2,1,1,1],[3,0,4,4,4,0,3,4,3,1,3,4,0,4,3,2],[1,0,0,0,0,0,0,1,2,4,4,4,0,1,0,0] >;
 

D12:S3 in GAP, Magma, Sage, TeX 

D_{12}\rtimes S_3
 
% in TeX
 
G:=Group("D12:S3");
 
// GroupNames label
 
G:=SmallGroup(144,139);
 
// by ID
 
G=gap.SmallGroup(144,139);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,50,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D12:S3 in TeX

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