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G = D4⋊D6order 96 = 25·3

2nd semidirect product of D4 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D44D6, Q85D6, C12.49D4, C12.17C23, D12.11C22, D4⋊S36C2, C4○D43S3, C3⋊C84C22, (C2×C6).8D4, C35(C8⋊C22), (C2×C4).22D6, C6.59(C2×D4), (C2×D12)⋊10C2, Q82S36C2, (C3×D4)⋊4C22, C4.Dic39C2, (C3×Q8)⋊4C22, C4.24(C3⋊D4), C4.17(C22×S3), (C2×C12).42C22, C22.5(C3⋊D4), (C3×C4○D4)⋊1C2, C2.23(C2×C3⋊D4), SmallGroup(96,156)

Series: Derived Chief Lower central Upper central

C1C12 — D4⋊D6
C1C3C6C12D12C2×D12 — D4⋊D6
C3C6C12 — D4⋊D6
C1C2C2×C4C4○D4

Generators and relations for D4⋊D6
 G = < a,b,c,d | a4=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >

Subgroups: 186 in 68 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C12 [×2], C12, D6 [×4], C2×C6, C2×C6, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3⋊C8 [×2], D12 [×2], D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C3×C4○D4, D4⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C3⋊D4 [×2], C22×S3, C8⋊C22, C2×C3⋊D4, D4⋊D6

Character table of D4⋊D6

 class 12A2B2C2D2E34A4B4C6A6B6C6D8A8B12A12B12C12D12E
 size 1124121222242444121222444
ρ1111111111111111111111    trivial
ρ2111-1-1-1111-111-1-11111-1-11    linear of order 2
ρ311-11-111-11-11-111-11-1-1-1-11    linear of order 2
ρ4111-111111-111-1-1-1-111-1-11    linear of order 2
ρ511-111-11-11-11-1111-1-1-1-1-11    linear of order 2
ρ611-1-11-11-1111-1-1-1-11-1-1111    linear of order 2
ρ71111-1-111111111-1-111111    linear of order 2
ρ811-1-1-111-1111-1-1-11-1-1-1111    linear of order 2
ρ9222200-1222-1-1-1-100-1-1-1-1-1    orthogonal lifted from S3
ρ1022-2-200-1-222-11110011-1-1-1    orthogonal lifted from D6
ρ1122-2200-1-22-2-11-1-1001111-1    orthogonal lifted from D6
ρ12222-200-122-2-1-11100-1-111-1    orthogonal lifted from D6
ρ132220002-2-20220000-2-200-2    orthogonal lifted from D4
ρ1422-200022-202-200002200-2    orthogonal lifted from D4
ρ1522-2000-12-20-11-3--300-1-1--3-31    complex lifted from C3⋊D4
ρ16222000-1-2-20-1-1-3--30011-3--31    complex lifted from C3⋊D4
ρ1722-2000-12-20-11--3-300-1-1-3--31    complex lifted from C3⋊D4
ρ18222000-1-2-20-1-1--3-30011--3-31    complex lifted from C3⋊D4
ρ194-400004000-40000000000    orthogonal lifted from C8⋊C22
ρ204-40000-2000200000-2323000    orthogonal faithful
ρ214-40000-200020000023-23000    orthogonal faithful

Permutation representations of D4⋊D6
On 24 points - transitive group 24T110
Generators in S24
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 19 16 22)(14 20 17 23)(15 21 18 24)
(1 22)(2 20)(3 24)(4 19)(5 23)(6 21)(7 13)(8 17)(9 15)(10 16)(11 14)(12 18)
(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 3)(4 6)(7 12)(8 11)(9 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

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

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

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

G:=TransitiveGroup(24,110);

D4⋊D6 is a maximal subgroup of
Q85D12  D4.10D12  Q8.8D12  Q8.9D12  M4(2).D6  M4(2).15D6  2+ 1+46S3  2- 1+44S3  D815D6  D811D6  S3×C8⋊C22  D24⋊C22  C12.C24  D12.32C23  D12.34C23  D4⋊D18  C12.4S4  D1220D6  D1218D6  Dic63D6  D126D6  C62.73D4  C12.7S4  D2021D6  D2019D6  Dic103D6  D20⋊D6  D4⋊D30
D4⋊D6 is a maximal quotient of
C4⋊C4.232D6  C4⋊C436D6  C4⋊C4.236D6  C12.38SD16  C42.48D6  C127D8  Q85Dic6  C42.56D6  Q82D12  C4⋊D4.S3  D1216D4  C4⋊D4⋊S3  (C2×C6).Q16  D12.36D4  C3⋊C86D4  C42.62D6  D12.23D4  C42.64D6  C42.68D6  D12.4Q8  C42.70D6  C4○D43Dic3  (C3×D4)⋊14D4  D4⋊D18  D1220D6  D1218D6  Dic63D6  D126D6  C62.73D4  D2021D6  D2019D6  Dic103D6  D20⋊D6  D4⋊D30

Matrix representation of D4⋊D6 in GL4(𝔽73) generated by

661400
59700
460759
0461466
,
2705928
0274514
759460
1466046
,
0100
72100
6043072
3030172
,
17200
07200
38656666
3035597
G:=sub<GL(4,GF(73))| [66,59,46,0,14,7,0,46,0,0,7,14,0,0,59,66],[27,0,7,14,0,27,59,66,59,45,46,0,28,14,0,46],[0,72,60,30,1,1,43,30,0,0,0,1,0,0,72,72],[1,0,38,30,72,72,65,35,0,0,66,59,0,0,66,7] >;

D4⋊D6 in GAP, Magma, Sage, TeX

D_4\rtimes D_6
% in TeX

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

G:=SmallGroup(96,156);
// by ID

G=gap.SmallGroup(96,156);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,188,579,159,69,2309]);
// Polycyclic

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

Export

Character table of D4⋊D6 in TeX

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