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## G = D6⋊D6order 144 = 24·32

### 2nd semidirect product of D6 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D6⋊D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — D6⋊D6
 Lower central C32 — C3×C6 — D6⋊D6
 Upper central C1 — C2 — C4

Generators and relations for D6⋊D6
G = < a,b,c,d | a6=b2=c6=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, dbd=a3b, dcd=c-1 >

Subgroups: 448 in 116 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×D4, D6⋊S3, C3×D12, C4×C3⋊S3, C2×S32, D6⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, S32, S3×D4, C2×S32, D6⋊D6

Character table of D6⋊D6

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D size 1 1 6 6 6 6 9 9 2 2 4 2 18 2 2 4 12 12 12 12 4 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 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 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 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 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 -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 -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 -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 -1 -1 -1 linear of order 2 ρ9 2 2 2 2 0 0 0 0 2 -1 -1 2 0 2 -1 -1 0 -1 -1 0 -1 2 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 2 0 0 0 0 2 -1 -1 -2 0 2 -1 -1 0 1 -1 0 1 -2 1 1 orthogonal lifted from D6 ρ11 2 -2 0 0 0 0 2 -2 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 0 0 2 2 0 0 -1 2 -1 2 0 -1 2 -1 -1 0 0 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 0 0 -2 2 0 0 -1 2 -1 -2 0 -1 2 -1 1 0 0 -1 -2 1 1 1 orthogonal lifted from D6 ρ14 2 -2 0 0 0 0 -2 2 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 0 0 2 -2 0 0 -1 2 -1 -2 0 -1 2 -1 -1 0 0 1 -2 1 1 1 orthogonal lifted from D6 ρ16 2 2 0 0 -2 -2 0 0 -1 2 -1 2 0 -1 2 -1 1 0 0 1 2 -1 -1 -1 orthogonal lifted from D6 ρ17 2 2 -2 -2 0 0 0 0 2 -1 -1 2 0 2 -1 -1 0 1 1 0 -1 2 -1 -1 orthogonal lifted from D6 ρ18 2 2 2 -2 0 0 0 0 2 -1 -1 -2 0 2 -1 -1 0 -1 1 0 1 -2 1 1 orthogonal lifted from D6 ρ19 4 4 0 0 0 0 0 0 -2 -2 1 -4 0 -2 -2 1 0 0 0 0 2 2 -1 -1 orthogonal lifted from C2×S32 ρ20 4 -4 0 0 0 0 0 0 4 -2 -2 0 0 -4 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ21 4 4 0 0 0 0 0 0 -2 -2 1 4 0 -2 -2 1 0 0 0 0 -2 -2 1 1 orthogonal lifted from S32 ρ22 4 -4 0 0 0 0 0 0 -2 4 -2 0 0 2 -4 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ23 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 0 0 3i -3i complex faithful ρ24 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 0 0 -3i 3i complex faithful

Permutation representations of D6⋊D6
On 24 points - transitive group 24T231
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)
(1 23 5 19 3 21)(2 22 6 24 4 20)(7 18 11 14 9 16)(8 17 12 13 10 15)
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 14)(8 15)(9 16)(10 17)(11 18)(12 13)

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

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

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

G:=TransitiveGroup(24,231);

Matrix representation of D6⋊D6 in GL4(𝔽5) generated by

 0 3 0 0 3 1 0 0 0 0 1 2 0 0 2 0
,
 0 0 0 3 0 0 3 1 1 2 0 0 2 0 0 0
,
 0 0 0 1 0 0 1 2 2 0 0 0 0 2 0 0
,
 0 0 2 0 0 0 4 3 3 0 0 0 1 2 0 0
G:=sub<GL(4,GF(5))| [0,3,0,0,3,1,0,0,0,0,1,2,0,0,2,0],[0,0,1,2,0,0,2,0,0,3,0,0,3,1,0,0],[0,0,2,0,0,0,0,2,0,1,0,0,1,2,0,0],[0,0,3,1,0,0,0,2,2,4,0,0,0,3,0,0] >;

D6⋊D6 in GAP, Magma, Sage, TeX

D_6\rtimes D_6
% in TeX

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

G:=SmallGroup(144,145);
// by ID

G=gap.SmallGroup(144,145);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,116,50,490,3461]);
// Polycyclic

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

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