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## G = Dic6⋊3D6order 288 = 25·32

### 3rd semidirect product of Dic6 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic6⋊3D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — Dic6⋊3D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6⋊3D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic63D6
G = < a,b,c,d | a12=c6=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a5, cbc-1=a6b, dbd=a9b, dcd=c-1 >

Subgroups: 706 in 147 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×5], C6 [×2], C6 [×6], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3, Dic3, C12 [×2], C12 [×3], D6, D6 [×8], C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6, C4×S3, D12, D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C3×D4 [×2], C3×D4 [×3], C3×Q8, C22×S3 [×2], C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S32 [×2], S3×C6, S3×C6, C2×C3⋊S3, C62, C8⋊S3, C24⋊C2, C4.Dic3, D4⋊S3, D4⋊S3 [×3], D4.S3, Q82S3 [×2], C3×D8, C2×D12, S3×D4, D42S3, C3×C4○D4, C3×C3⋊C8, C324C8, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C12⋊S3, D4×C32, C2×S32, D8⋊S3, D4⋊D6, D6.Dic3, Dic6⋊S3, C325SD16, C3×D4⋊S3, C327D8, S3×D12, C3×D42S3, Dic63D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C8⋊C22, S32, S3×D4, C2×C3⋊D4, C2×S32, D8⋊S3, D4⋊D6, S3×C3⋊D4, Dic63D6

Smallest permutation representation of Dic63D6
On 48 points
Generators in S48
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 43 7 37)(2 42 8 48)(3 41 9 47)(4 40 10 46)(5 39 11 45)(6 38 12 44)(13 28 19 34)(14 27 20 33)(15 26 21 32)(16 25 22 31)(17 36 23 30)(18 35 24 29)
(1 31 9 35 5 27)(2 36 10 28 6 32)(3 29 11 33 7 25)(4 34 12 26 8 30)(13 38 21 42 17 46)(14 43 22 47 18 39)(15 48 23 40 19 44)(16 41 24 45 20 37)
(1 9)(2 8)(3 7)(4 6)(10 12)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(26 36)(27 35)(28 34)(29 33)(30 32)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)

G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,43,7,37)(2,42,8,48)(3,41,9,47)(4,40,10,46)(5,39,11,45)(6,38,12,44)(13,28,19,34)(14,27,20,33)(15,26,21,32)(16,25,22,31)(17,36,23,30)(18,35,24,29), (1,31,9,35,5,27)(2,36,10,28,6,32)(3,29,11,33,7,25)(4,34,12,26,8,30)(13,38,21,42,17,46)(14,43,22,47,18,39)(15,48,23,40,19,44)(16,41,24,45,20,37), (1,9)(2,8)(3,7)(4,6)(10,12)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(26,36)(27,35)(28,34)(29,33)(30,32)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)>;

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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,43,7,37)(2,42,8,48)(3,41,9,47)(4,40,10,46)(5,39,11,45)(6,38,12,44)(13,28,19,34)(14,27,20,33)(15,26,21,32)(16,25,22,31)(17,36,23,30)(18,35,24,29), (1,31,9,35,5,27)(2,36,10,28,6,32)(3,29,11,33,7,25)(4,34,12,26,8,30)(13,38,21,42,17,46)(14,43,22,47,18,39)(15,48,23,40,19,44)(16,41,24,45,20,37), (1,9)(2,8)(3,7)(4,6)(10,12)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(26,36)(27,35)(28,34)(29,33)(30,32)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47) );

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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,43,7,37),(2,42,8,48),(3,41,9,47),(4,40,10,46),(5,39,11,45),(6,38,12,44),(13,28,19,34),(14,27,20,33),(15,26,21,32),(16,25,22,31),(17,36,23,30),(18,35,24,29)], [(1,31,9,35,5,27),(2,36,10,28,6,32),(3,29,11,33,7,25),(4,34,12,26,8,30),(13,38,21,42,17,46),(14,43,22,47,18,39),(15,48,23,40,19,44),(16,41,24,45,20,37)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(26,36),(27,35),(28,34),(29,33),(30,32),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47)])

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 12A 12B 12C 12D 12E 12F 12G 24A 24B order 1 2 2 2 2 2 3 3 3 4 4 4 6 6 6 6 6 6 6 6 6 6 8 8 12 12 12 12 12 12 12 24 24 size 1 1 4 6 12 36 2 2 4 2 6 12 2 2 4 4 4 8 8 8 12 24 12 36 4 4 6 6 8 12 12 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C8⋊C22 S32 S3×D4 C2×S32 D8⋊S3 D4⋊D6 S3×C3⋊D4 Dic6⋊3D6 kernel Dic6⋊3D6 D6.Dic3 Dic6⋊S3 C32⋊5SD16 C3×D4⋊S3 C32⋊7D8 S3×D12 C3×D4⋊2S3 D4⋊S3 D4⋊2S3 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 D12 C3×D4 Dic3 D6 C32 D4 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 2 2 2 1

Matrix representation of Dic63D6 in GL8(𝔽73)

 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 72 72 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 72 1 2 0 0 0 0 0 1 72 72
,
 1 1 2 1 0 0 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 23 50 57 0 0 0 0 0 34 34 16 32 0 0 0 0 39 23 27 27 0 0 0 0 34 11 23 62
,
 1 1 1 2 0 0 0 0 1 1 2 1 0 0 0 0 0 72 72 72 0 0 0 0 72 72 72 72 0 0 0 0 0 0 0 0 39 23 27 27 0 0 0 0 23 39 0 27 0 0 0 0 23 50 57 0 0 0 0 0 39 62 50 11
,
 72 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 72 72 72 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 72 1 2 0 0 0 0 1 0 0 72

G:=sub<GL(8,GF(73))| [72,72,1,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,72,1,0,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72],[1,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,2,1,72,72,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,23,34,39,34,0,0,0,0,50,34,23,11,0,0,0,0,57,16,27,23,0,0,0,0,0,32,27,62],[1,1,0,72,0,0,0,0,1,1,72,72,0,0,0,0,1,2,72,72,0,0,0,0,2,1,72,72,0,0,0,0,0,0,0,0,39,23,23,39,0,0,0,0,23,39,50,62,0,0,0,0,27,0,57,50,0,0,0,0,27,27,0,11],[72,72,0,1,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,0,72,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,72] >;

Dic63D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_3D_6
% in TeX

G:=Group("Dic6:3D6");
// GroupNames label

G:=SmallGroup(288,573);
// by ID

G=gap.SmallGroup(288,573);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,346,185,80,1356,9414]);
// 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^5,c*b*c^-1=a^6*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations

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