Copied to
clipboard

G = D46D6order 96 = 25·3

2nd semidirect product of D4 and D6 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D46D6, C233D6, C6.7C24, D128C22, D6.3C23, C312+ 1+4, C12.21C23, Dic68C22, Dic3.4C23, (C2×C4)⋊3D6, (C2×D4)⋊7S3, (S3×D4)⋊4C2, (C6×D4)⋊7C2, D4(C3⋊D4), C4○D125C2, D42S34C2, (C4×S3)⋊1C22, (C2×C12)⋊3C22, (C3×D4)⋊7C22, C3⋊D43C22, (C2×C6).2C23, C2.8(S3×C23), C4.21(C22×S3), (C22×C6)⋊5C22, (C22×S3)⋊3C22, (C2×Dic3)⋊4C22, C22.6(C22×S3), (C2×C3⋊D4)⋊11C2, SmallGroup(96,211)

Series: Derived Chief Lower central Upper central

C1C6 — D46D6
C1C3C6D6C22×S3S3×D4 — D46D6
C3C6 — D46D6
C1C2C2×D4

Generators and relations for D46D6
 G = < a,b,c,d | a4=b2=c6=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 370 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], S3 [×4], C6, C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], Dic3 [×4], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×C6 [×4], C2×C6 [×2], C2×D4, C2×D4 [×8], C4○D4 [×6], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×2], 2+ 1+4, C4○D12 [×2], S3×D4 [×4], D42S3 [×4], C2×C3⋊D4 [×4], C6×D4, D46D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ 1+4, S3×C23, D46D6

Character table of D46D6

 class 12A2B2C2D2E2F2G2H2I2J34A4B4C4D4E4F6A6B6C6D6E6F6G12A12B
 size 112222266662226666222444444
ρ1111111111111111111111111111    trivial
ρ211-1-1-1-11-1-111111-1-111111-1-1-1-111    linear of order 2
ρ311-1-1-1-1111-1-111111-1-1111-1-1-1-111    linear of order 2
ρ41111111-1-1-1-1111-1-1-1-1111111111    linear of order 2
ρ511-11-11-11-1-1-11-11-1111-1-1111-1-11-1    linear of order 2
ρ6111-11-1-1-11-1-11-111-111-1-11-1-1111-1    linear of order 2
ρ711-11-11-1-11111-111-1-1-1-1-1111-1-11-1    linear of order 2
ρ8111-11-1-11-1111-11-11-1-1-1-11-1-1111-1    linear of order 2
ρ91111-1-1-1-1-1-1111-1111-1-1-111-1-11-11    linear of order 2
ρ1011-1-111-111-1111-1-1-11-1-1-11-111-1-11    linear of order 2
ρ111111-1-1-1111-111-1-1-1-11-1-111-1-11-11    linear of order 2
ρ1211-1-111-1-1-11-111-111-11-1-11-111-1-11    linear of order 2
ρ13111-1-1111-11-11-1-11-11-1111-11-11-1-1    linear of order 2
ρ1411-111-11-111-11-1-1-111-11111-11-1-1-1    linear of order 2
ρ15111-1-111-11-111-1-1-11-11111-11-11-1-1    linear of order 2
ρ1611-111-111-1-111-1-11-1-111111-11-1-1-1    linear of order 2
ρ172222-2-2-20000-12-2000011-1-111-11-1    orthogonal lifted from D6
ρ18222-2-2220000-1-2-20000-1-1-11-11-111    orthogonal lifted from D6
ρ1922-222-220000-1-2-20000-1-1-1-11-1111    orthogonal lifted from D6
ρ2022-2-222-20000-12-2000011-11-1-111-1    orthogonal lifted from D6
ρ21222-22-2-20000-1-22000011-111-1-1-11    orthogonal lifted from D6
ρ2222222220000-1220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ2322-2-2-2-220000-1220000-1-1-11111-1-1    orthogonal lifted from D6
ρ2422-22-22-20000-1-22000011-1-1-111-11    orthogonal lifted from D6
ρ254-4000000000400000000-4000000    orthogonal lifted from 2+ 1+4
ρ264-4000000000-20000002-3-2-32000000    complex faithful
ρ274-4000000000-2000000-2-32-32000000    complex faithful

Permutation representations of D46D6
On 24 points - transitive group 24T95
Generators in S24
(1 24 8 21)(2 22 9 19)(3 20 7 23)(4 15 10 18)(5 13 11 16)(6 17 12 14)
(1 21)(2 19)(3 23)(4 15)(5 13)(6 17)(7 20)(8 24)(9 22)(10 18)(11 16)(12 14)
(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 5)(2 4)(3 6)(7 12)(8 11)(9 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

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

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

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

G:=TransitiveGroup(24,95);

On 24 points - transitive group 24T100
Generators in S24
(1 22 14 9)(2 10 15 23)(3 24 16 11)(4 12 17 19)(5 20 18 7)(6 8 13 21)
(1 9)(2 23)(3 11)(4 19)(5 7)(6 21)(8 13)(10 15)(12 17)(14 22)(16 24)(18 20)
(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 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)

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

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

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

G:=TransitiveGroup(24,100);

D46D6 is a maximal subgroup of
C23⋊D12  C23.5D12  M4(2)⋊D6  D121D4  C246D6  C22⋊C4⋊D6  C427D6  C428D6  D813D6  SD1613D6  D85D6  D86D6  C6.C25  S3×2+ 1+4  D6.C24  D46D18  D1224D6  D1212D6  D1213D6  C32⋊2+ 1+4  C3282+ 1+4  D2025D6  D2013D6  D2014D6  C15⋊2+ 1+4  D46D30
D46D6 is a maximal quotient of
C233Dic6  C24.35D6  C24.38D6  C234D12  C24.41D6  C24.42D6  C6.72+ 1+4  C6.82+ 1+4  C6.2+ 1+4  C6.102+ 1+4  C6.112+ 1+4  C6.62- 1+4  D45Dic6  C42.104D6  C4213D6  C42.108D6  D45D12  C4218D6  C42.113D6  C42.114D6  C4219D6  C42.115D6  C42.116D6  C42.118D6  C24.43D6  C247D6  C248D6  C24.44D6  C24.45D6  C24.46D6  C249D6  C24.47D6  C6.322+ 1+4  Dic620D4  C6.342+ 1+4  C6.702- 1+4  C6.712- 1+4  C6.372+ 1+4  C6.382+ 1+4  C6.722- 1+4  C6.402+ 1+4  D1220D4  C6.422+ 1+4  C6.432+ 1+4  C6.442+ 1+4  C6.452+ 1+4  C6.462+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C6.492+ 1+4  C6.752- 1+4  C6.512+ 1+4  C6.522+ 1+4  C6.532+ 1+4  C6.202- 1+4  C6.222- 1+4  C6.562+ 1+4  C6.782- 1+4  C6.252- 1+4  C6.592+ 1+4  C6.812- 1+4  C6.612+ 1+4  C6.622+ 1+4  C6.632+ 1+4  C6.642+ 1+4  C6.652+ 1+4  C6.662+ 1+4  C6.672+ 1+4  C6.682+ 1+4  C6.692+ 1+4  C42.137D6  C42.138D6  C42.140D6  C4222D6  C4223D6  C4224D6  C42.145D6  C42.166D6  C4228D6  D1211D4  Dic611D4  C42.168D6  C4230D6  Dic69Q8  C42.174D6  D129Q8  C42.178D6  C42.179D6  C42.180D6  C24.49D6  D4×C3⋊D4  C2412D6  C24.52D6  C24.53D6  D46D18  D1224D6  D1212D6  D1213D6  C32⋊2+ 1+4  C3282+ 1+4  D2025D6  D2013D6  D2014D6  C15⋊2+ 1+4  D46D30

Matrix representation of D46D6 in GL4(𝔽7) generated by

3100
4400
4563
0341
,
6656
4013
1165
1632
,
4415
5465
6463
4450
,
6611
0601
0511
0001
G:=sub<GL(4,GF(7))| [3,4,4,0,1,4,5,3,0,0,6,4,0,0,3,1],[6,4,1,1,6,0,1,6,5,1,6,3,6,3,5,2],[4,5,6,4,4,4,4,4,1,6,6,5,5,5,3,0],[6,0,0,0,6,6,5,0,1,0,1,0,1,1,1,1] >;

D46D6 in GAP, Magma, Sage, TeX

D_4\rtimes_6D_6
% in TeX

G:=Group("D4:6D6");
// GroupNames label

G:=SmallGroup(96,211);
// by ID

G=gap.SmallGroup(96,211);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,579,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D46D6 in TeX

׿
×
𝔽