Copied to
clipboard

G = D6.3D6order 144 = 24·32

3rd non-split extension by D6 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D6.3D6, Dic3.3D6, C62.7C22, C3⋊D43S3, C22.1S32, (C2×C6).4D6, C3⋊D122C2, C34(C4○D12), (C6×Dic3)⋊6C2, (C2×Dic3)⋊3S3, (S3×Dic3)⋊5C2, C322Q84C2, C6.D62C2, C325(C4○D4), C327D41C2, C33(D42S3), (S3×C6).3C22, C6.11(C22×S3), (C3×C6).11C23, C3⋊Dic3.6C22, (C3×Dic3).9C22, C2.12(C2×S32), (C3×C3⋊D4)⋊1C2, (C2×C3⋊S3).5C22, SmallGroup(144,147)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D6.3D6
C1C3C32C3×C6S3×C6S3×Dic3 — D6.3D6
C32C3×C6 — D6.3D6
C1C2C22

Generators and relations for D6.3D6
 G = < a,b,c,d | a6=b2=c6=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 284 in 88 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×4], C22, C22 [×2], S3 [×4], C6 [×2], C6 [×5], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×3], Dic3 [×3], C12 [×3], D6, D6 [×3], C2×C6 [×2], C2×C6 [×2], C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6 [×2], C4×S3 [×3], D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4 [×4], C2×C12, C3×D4, C3×Dic3 [×3], C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C4○D12, D42S3, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C6×Dic3, C3×C3⋊D4, C327D4, D6.3D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, C4○D12, D42S3, C2×S32, D6.3D6

Character table of D6.3D6

 class 12A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I12A12B12C12D12E
 size 1126182243366182222444412666612
ρ1111111111111111111111111111    trivial
ρ211-111111-1-11-1-1-111-1-1-1-111-11-11-1    linear of order 2
ρ3111-1-1111111-1-111111111-11111-1    linear of order 2
ρ411-1-1-1111-1-1111-111-1-1-1-11-1-11-111    linear of order 2
ρ5111-11111-1-1-1-1111111111-1-1-1-1-1-1    linear of order 2
ρ611-1-1111111-11-1-111-1-1-1-11-11-11-11    linear of order 2
ρ71111-1111-1-1-11-1111111111-1-1-1-11    linear of order 2
ρ811-11-111111-1-11-111-1-1-1-1111-11-1-1    linear of order 2
ρ922-200-12-122-2001-121-211-10-11-110    orthogonal lifted from D6
ρ1022200-12-122200-1-12-12-1-1-10-1-1-1-10    orthogonal lifted from S3
ρ1122-2-202-1-100020-22-1-2111-110000-1    orthogonal lifted from D6
ρ12222202-1-10002022-12-1-1-1-1-10000-1    orthogonal lifted from S3
ρ1322-200-12-1-2-22001-121-211-101-11-10    orthogonal lifted from D6
ρ14222-202-1-1000-2022-12-1-1-1-1100001    orthogonal lifted from D6
ρ1522-2202-1-1000-20-22-1-2111-1-100001    orthogonal lifted from D6
ρ1622200-12-1-2-2-200-1-12-12-1-1-1011110    orthogonal lifted from D6
ρ172-20002222i-2i0000-2-20000-202i0-2i00    complex lifted from C4○D4
ρ182-2000222-2i2i0000-2-20000-20-2i02i00    complex lifted from C4○D4
ρ192-2000-12-12i-2i000--31-2-30-3--310-i-3i30    complex lifted from C4○D12
ρ202-2000-12-1-2i2i000-31-2--30--3-310i-3-i30    complex lifted from C4○D12
ρ212-2000-12-1-2i2i000--31-2-30-3--310i3-i-30    complex lifted from C4○D12
ρ222-2000-12-12i-2i000-31-2--30--3-310-i3i-30    complex lifted from C4○D12
ρ2344-400-2-21000002-2-222-1-11000000    orthogonal lifted from C2×S32
ρ2444400-2-2100000-2-2-2-2-2111000000    orthogonal lifted from S32
ρ254-40004-2-2000000-4200002000000    symplectic lifted from D42S3, Schur index 2
ρ264-4000-2-2100000-2-3222-30--3-3-1000000    complex faithful
ρ274-4000-2-21000002-322-2-30-3--3-1000000    complex faithful

Permutation representations of D6.3D6
On 24 points - transitive group 24T205
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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 12 4 9)(2 7 5 10)(3 8 6 11)(13 23 16 20)(14 24 17 21)(15 19 18 22)

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

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

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

G:=TransitiveGroup(24,205);

On 24 points - transitive group 24T221
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)
(1 9 5 7 3 11)(2 10 6 8 4 12)(13 23 15 19 17 21)(14 24 16 20 18 22)
(1 24 4 21)(2 19 5 22)(3 20 6 23)(7 18 10 15)(8 13 11 16)(9 14 12 17)

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

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

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

G:=TransitiveGroup(24,221);

D6.3D6 is a maximal subgroup of
D12.33D6  S3×C4○D12  Dic6.24D6  S3×D42S3  Dic612D6  D1213D6  C32⋊2+ 1+4  D18.3D6  Dic3.D18  C62.8D6  C62.9D6  (S3×C6).D6  D6.S32  D6.4S32  D6.3S32  C62.90D6  C62.93D6  C62.96D6
D6.3D6 is a maximal quotient of
C62.6C23  Dic35Dic6  C62.16C23  C62.17C23  C62.18C23  Dic3.D12  C62.24C23  C62.28C23  C62.29C23  C62.37C23  C62.38C23  C62.47C23  Dic34D12  D6.D12  C62.65C23  D64Dic6  C62.94C23  C62.95C23  C62.97C23  C62.98C23  C62.100C23  C62.101C23  C62.56D4  C623Q8  C62.60D4  C62.111C23  C62.113C23  Dic3×C3⋊D4  C62.117C23  C626D4  D18.3D6  Dic3.D18  C62.8D6  (S3×C6).D6  D6.S32  D6.4S32  D6.3S32  C62.90D6  C62.93D6  C62.96D6

Matrix representation of D6.3D6 in GL4(𝔽7) generated by

0031
5460
1655
3320
,
3315
2520
6612
3155
,
1113
0663
2264
2621
,
0013
4014
3351
1632
G:=sub<GL(4,GF(7))| [0,5,1,3,0,4,6,3,3,6,5,2,1,0,5,0],[3,2,6,3,3,5,6,1,1,2,1,5,5,0,2,5],[1,0,2,2,1,6,2,6,1,6,6,2,3,3,4,1],[0,4,3,1,0,0,3,6,1,1,5,3,3,4,1,2] >;

D6.3D6 in GAP, Magma, Sage, TeX

D_6._3D_6
% in TeX

G:=Group("D6.3D6");
// GroupNames label

G:=SmallGroup(144,147);
// by ID

G=gap.SmallGroup(144,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^6=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D6.3D6 in TeX

׿
×
𝔽