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## G = Dic3⋊6S32order 432 = 24·33

### 2nd semidirect product of Dic3 and S32 acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — Dic3⋊6S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C3×C6.D6 — Dic3⋊6S32
 Lower central C33 — Dic3⋊6S32
 Upper central C1 — C2

Generators and relations for Dic36S32
G = < a,b,c,d,e,f | a6=c3=d2=e3=f2=1, b2=a3, bab-1=dad=faf=a-1, ac=ca, ae=ea, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1452 in 270 conjugacy classes, 62 normal (7 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, C32×Dic3, C335C4, C324D6, C6×C3⋊S3, C4×S32, C3×C6.D6, Dic3×C3⋊S3, C2×C324D6, Dic36S32
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S32, S3×C2×C4, C2×S32, C4×S32, S33, Dic36S32

Smallest permutation representation of Dic36S32
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 28 4 25)(2 27 5 30)(3 26 6 29)(7 32 10 35)(8 31 11 34)(9 36 12 33)(13 38 16 41)(14 37 17 40)(15 42 18 39)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 34)(2 33)(3 32)(4 31)(5 36)(6 35)(7 29)(8 28)(9 27)(10 26)(11 25)(12 30)(13 44)(14 43)(15 48)(16 47)(17 46)(18 45)(19 41)(20 40)(21 39)(22 38)(23 37)(24 42)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 37)(2 42)(3 41)(4 40)(5 39)(6 38)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(13 26)(14 25)(15 30)(16 29)(17 28)(18 27)(19 32)(20 31)(21 36)(22 35)(23 34)(24 33)

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

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

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

54 conjugacy classes

 class 1 2A 2B ··· 2G 3A 3B 3C 3D 3E 3F 3G 4A ··· 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 6H ··· 6M 12A ··· 12L 12M ··· 12R order 1 2 2 ··· 2 3 3 3 3 3 3 3 4 ··· 4 4 4 6 6 6 6 6 6 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 9 ··· 9 2 2 2 4 4 4 8 3 ··· 3 27 27 2 2 2 4 4 4 8 18 ··· 18 6 ··· 6 12 ··· 12

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + - image C1 C2 C2 C2 C4 S3 D6 D6 C4×S3 S32 C2×S32 C4×S32 S33 Dic3⋊6S32 kernel Dic3⋊6S32 C3×C6.D6 Dic3×C3⋊S3 C2×C32⋊4D6 C32⋊4D6 C6.D6 C3×Dic3 C2×C3⋊S3 C3⋊S3 Dic3 C6 C3 C2 C1 # reps 1 3 3 1 8 3 6 3 12 3 3 6 1 1

Matrix representation of Dic36S32 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 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 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 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 12 12
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12

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

Dic36S32 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_6S_3^2
% in TeX

G:=Group("Dic3:6S3^2");
// GroupNames label

G:=SmallGroup(432,596);
// by ID

G=gap.SmallGroup(432,596);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,58,298,2028,14118]);
// Polycyclic

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

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