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G = D12⋊S3order 144 = 24·32

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

metabelian, supersoluble, monomial

Aliases: D123S3, D6.2D6, Dic63S3, C12.22D6, Dic3.1D6, C4.11S32, (C3×D12)⋊5C2, C3⋊D123C2, (C3×Dic6)⋊5C2, (S3×Dic3)⋊1C2, C322(C4○D4), (C3×C6).3C23, C6.3(C22×S3), C31(D42S3), C32(Q83S3), (S3×C6).2C22, (C3×C12).18C22, C3⋊Dic3.11C22, (C3×Dic3).2C22, C2.6(C2×S32), (C4×C3⋊S3)⋊1C2, (C2×C3⋊S3).11C22, SmallGroup(144,139)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D12⋊S3
C1C3C32C3×C6S3×C6S3×Dic3 — D12⋊S3
C32C3×C6 — D12⋊S3
C1C2C4

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 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×3], C2×C6 [×2], C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, Dic6, C4×S3 [×5], D12, D12 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, C3×Q8, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, D42S3, Q83S3, S3×Dic3 [×2], C3⋊D12 [×2], C3×Dic6, C3×D12, C4×C3⋊S3, D12⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, D42S3, Q83S3, C2×S32, 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 C4○D4
ρ182-20002220002i-2i-2-2-200000000    complex lifted from C4○D4
ρ1944000-2-21-40000-2-210022-1-100    orthogonal lifted from C2×S32
ρ204-4000-24-2000002-4200000000    orthogonal lifted from Q83S3, Schur index 2
ρ2144000-2-2140000-2-2100-2-21100    orthogonal lifted from S32
ρ224-40004-2-200000-42200000000    symplectic lifted from D42S3, 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
C246D6  D12.2D6  D245S3  D12.4D6  D12.8D6  D125D6  D12.14D6  D12.15D6  D12.33D6  D1223D6  D1224D6  S3×D42S3  D1213D6  D12.25D6  S3×Q83S3  D18.D6  D12⋊D9  C12⋊S3⋊S3  C12.S32  D6.3S32  D6.6S32  D12⋊(C3⋊S3)  C12.39S32  C12⋊S312S3
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  D61Dic6  D6.D12  D12⋊Dic3  C62.77C23  C122D12  D18.D6  D12⋊D9  C12.84S32  D6.3S32  D6.6S32  D12⋊(C3⋊S3)  C12.39S32  C12⋊S312S3

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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