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## G = Dic6⋊D6order 288 = 25·32

### 4th semidirect product of Dic6 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic6⋊D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — Dic3.D6 — Dic6⋊D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6⋊D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic6⋊D6
G = < a,b,c,d | a12=c6=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a7, cbc-1=dbd=a9b, dcd=c-1 >

Subgroups: 834 in 163 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×10], C6 [×2], C6 [×5], C8 [×2], C2×C4 [×2], D4, D4 [×2], Q8 [×3], C23, C32, Dic3 [×5], C12 [×2], C12 [×3], D6 [×15], C2×C6 [×4], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×2], C4×S3 [×5], D12 [×3], C3⋊D4 [×4], C3×D4 [×2], C3×D4, C3×Q8 [×2], C22×S3 [×4], C2×SD16, C3×Dic3 [×2], C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3 [×3], C62, S3×C8 [×2], C24⋊C2 [×2], D4.S3 [×2], Q82S3 [×2], C3×SD16 [×2], S3×D4 [×3], S3×Q8 [×2], C3×C3⋊C8 [×2], C6.D6, C322Q8, C3×Dic6 [×2], C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C22×C3⋊S3, S3×SD16 [×2], C12.29D6, C325SD16 [×2], C3×D4.S3 [×2], Dic3.D6, D4×C3⋊S3, Dic6⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, C22×S3 [×2], C2×SD16, S32, S3×D4 [×2], C2×S32, S3×SD16 [×2], Dic3⋊D6, Dic6⋊D6

Permutation representations of Dic6⋊D6
On 24 points - transitive group 24T670
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 21 7 15)(2 20 8 14)(3 19 9 13)(4 18 10 24)(5 17 11 23)(6 16 12 22)
(1 5 9)(2 12 10 8 6 4)(3 7 11)(13 24 17 16 21 20)(14 19 18 23 22 15)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 18)(14 17)(15 16)(19 24)(20 23)(21 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), (1,21,7,15)(2,20,8,14)(3,19,9,13)(4,18,10,24)(5,17,11,23)(6,16,12,22), (1,5,9)(2,12,10,8,6,4)(3,7,11)(13,24,17,16,21,20)(14,19,18,23,22,15), (1,3)(4,12)(5,11)(6,10)(7,9)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)>;

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

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

G:=TransitiveGroup(24,670);

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E 24A 24B 24C 24D order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 24 24 24 24 size 1 1 4 9 9 36 2 2 4 2 12 12 18 2 2 4 8 8 8 8 6 6 6 6 4 4 8 24 24 12 12 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 SD16 S32 S3×D4 C2×S32 S3×SD16 Dic3⋊D6 Dic6⋊D6 kernel Dic6⋊D6 C12.29D6 C32⋊5SD16 C3×D4.S3 Dic3.D6 D4×C3⋊S3 D4.S3 C3⋊Dic3 C2×C3⋊S3 C3⋊C8 Dic6 C3×D4 C3⋊S3 D4 C6 C4 C3 C2 C1 # reps 1 1 2 2 1 1 2 1 1 2 2 2 4 1 2 1 4 2 1

Matrix representation of Dic6⋊D6 in GL6(𝔽73)

 72 48 0 0 0 0 3 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 12 4 0 0 0 0 55 61 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 70 72 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 3 1 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [72,3,0,0,0,0,48,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[12,55,0,0,0,0,4,61,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,70,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,3,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic6⋊D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes D_6
% in TeX

G:=Group("Dic6:D6");
// GroupNames label

G:=SmallGroup(288,578);
// by ID

G=gap.SmallGroup(288,578);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^6=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations

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