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## G = D4⋊D6order 96 = 25·3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D4⋊D6
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — D4⋊D6
 Lower central C3 — C6 — C12 — D4⋊D6
 Upper central C1 — C2 — C2×C4 — C4○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 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E size 1 1 2 4 12 12 2 2 2 4 2 4 4 4 12 12 2 2 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 linear of order 2 ρ3 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ5 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 linear of order 2 ρ6 1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 linear of order 2 ρ7 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 linear of order 2 ρ8 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 linear of order 2 ρ9 2 2 2 2 0 0 -1 2 2 2 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 0 0 -1 -2 2 2 -1 1 1 1 0 0 1 1 -1 -1 -1 orthogonal lifted from D6 ρ11 2 2 -2 2 0 0 -1 -2 2 -2 -1 1 -1 -1 0 0 1 1 1 1 -1 orthogonal lifted from D6 ρ12 2 2 2 -2 0 0 -1 2 2 -2 -1 -1 1 1 0 0 -1 -1 1 1 -1 orthogonal lifted from D6 ρ13 2 2 2 0 0 0 2 -2 -2 0 2 2 0 0 0 0 -2 -2 0 0 -2 orthogonal lifted from D4 ρ14 2 2 -2 0 0 0 2 2 -2 0 2 -2 0 0 0 0 2 2 0 0 -2 orthogonal lifted from D4 ρ15 2 2 -2 0 0 0 -1 2 -2 0 -1 1 √-3 -√-3 0 0 -1 -1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ16 2 2 2 0 0 0 -1 -2 -2 0 -1 -1 √-3 -√-3 0 0 1 1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ17 2 2 -2 0 0 0 -1 2 -2 0 -1 1 -√-3 √-3 0 0 -1 -1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ18 2 2 2 0 0 0 -1 -2 -2 0 -1 -1 -√-3 √-3 0 0 1 1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 -2√3 2√3 0 0 0 orthogonal faithful ρ21 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 2√3 -2√3 0 0 0 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);

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

 66 14 0 0 59 7 0 0 46 0 7 59 0 46 14 66
,
 27 0 59 28 0 27 45 14 7 59 46 0 14 66 0 46
,
 0 1 0 0 72 1 0 0 60 43 0 72 30 30 1 72
,
 1 72 0 0 0 72 0 0 38 65 66 66 30 35 59 7
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

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