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## G = Dic6⋊12D6order 288 = 25·32

### 6th semidirect product of Dic6 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic6⋊12D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — Dic6⋊12D6
 Lower central C32 — C3×C6 — Dic6⋊12D6
 Upper central C1 — C2 — D4

Generators and relations for Dic612D6
G = < a,b,c,d | a12=c6=d2=1, b2=a6, bab-1=a-1, cac-1=a7, dad=a5, cbc-1=a6b, bd=db, dcd=c-1 >

Subgroups: 1346 in 355 conjugacy classes, 110 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C2×C4○D4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C4○D12, S3×D4, D42S3, D42S3, S3×Q8, Q83S3, C3×C4○D4, S3×Dic3, C6.D6, C6.D6, C3⋊D12, C322Q8, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C2×S32, C22×C3⋊S3, S3×C4○D4, Dic3.D6, D6.6D6, C4×S32, D6.3D6, C2×C6.D6, Dic3⋊D6, C3×D42S3, D4×C3⋊S3, Dic612D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, S3×C23, C2×S32, S3×C4○D4, C22×S32, Dic612D6

Permutation representations of Dic612D6
On 24 points - transitive group 24T608
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 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)
(1 2 9 10 5 6)(3 4 11 12 7 8)(13 24 17 16 21 20)(14 19 18 23 22 15)
(1 3)(2 8)(4 6)(5 11)(7 9)(10 12)(13 23)(14 16)(15 21)(17 19)(18 24)(20 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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17), (1,2,9,10,5,6)(3,4,11,12,7,8)(13,24,17,16,21,20)(14,19,18,23,22,15), (1,3)(2,8)(4,6)(5,11)(7,9)(10,12)(13,23)(14,16)(15,21)(17,19)(18,24)(20,22)>;

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

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

G:=TransitiveGroup(24,608);

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C ··· 6G 6H 6I 6J 6K 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K order 1 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 12 12 12 12 12 12 size 1 1 2 2 6 6 9 9 18 18 2 2 4 2 3 3 3 3 6 6 6 6 18 2 2 4 ··· 4 8 8 12 12 4 4 6 6 6 6 8 12 12 12 12

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 D6 C4○D4 S32 C2×S32 C2×S32 S3×C4○D4 Dic6⋊12D6 kernel Dic6⋊12D6 Dic3.D6 D6.6D6 C4×S32 D6.3D6 C2×C6.D6 Dic3⋊D6 C3×D4⋊2S3 D4×C3⋊S3 D4⋊2S3 Dic6 C4×S3 C2×Dic3 C3⋊D4 C3×D4 C3⋊S3 D4 C4 C22 C3 C1 # reps 1 1 2 1 4 2 2 2 1 2 2 2 4 4 2 4 1 1 2 4 1

Matrix representation of Dic612D6 in GL6(𝔽13)

 12 5 0 0 0 0 10 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 2 8 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 8 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 12 1

G:=sub<GL(6,GF(13))| [12,10,0,0,0,0,5,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,2,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;

Dic612D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{12}D_6
% in TeX

G:=Group("Dic6:12D6");
// GroupNames label

G:=SmallGroup(288,960);
// by ID

G=gap.SmallGroup(288,960);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,346,185,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^6=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,d*a*d=a^5,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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