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## G = Dic3×C32⋊C4order 432 = 24·33

### Direct product of Dic3 and C32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — Dic3×C32⋊C4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — Dic3×C3⋊S3 — Dic3×C32⋊C4
 Lower central C33 — Dic3×C32⋊C4
 Upper central C1 — C2

Generators and relations for Dic3×C32⋊C4
G = < a,b,c,d,e | a6=c3=d3=e4=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 704 in 104 conjugacy classes, 28 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C42, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, C32⋊C4, S3×C6, C2×C3⋊S3, C4×Dic3, C3×C3⋊S3, C32×C6, S3×Dic3, C4×C3⋊S3, C2×C32⋊C4, C2×C32⋊C4, C32×Dic3, C335C4, C3×C32⋊C4, C33⋊C4, C6×C3⋊S3, C4×C32⋊C4, Dic3×C3⋊S3, C6×C32⋊C4, C2×C33⋊C4, Dic3×C32⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C4×S3, C2×Dic3, C32⋊C4, C4×Dic3, C2×C32⋊C4, C4×C32⋊C4, S3×C32⋊C4, Dic3×C32⋊C4

Smallest permutation representation of Dic3×C32⋊C4
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 3 5)(2 4 6)(7 11 9)(8 12 10)(25 29 27)(26 30 28)(31 33 35)(32 34 36)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 45 47)(44 46 48)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 8 type + + + + + - + + + + - image C1 C2 C2 C2 C4 C4 C4 C4 S3 Dic3 D6 C4×S3 C4×S3 C32⋊C4 C2×C32⋊C4 C4×C32⋊C4 S3×C32⋊C4 Dic3×C32⋊C4 kernel Dic3×C32⋊C4 Dic3×C3⋊S3 C6×C32⋊C4 C2×C33⋊C4 C32×Dic3 C33⋊5C4 C3×C32⋊C4 C33⋊C4 C2×C32⋊C4 C32⋊C4 C2×C3⋊S3 C3⋊S3 C3×C6 Dic3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 4 4 1 2 1 2 2 2 2 4 2 2

Matrix representation of Dic3×C32⋊C4 in GL6(𝔽13)

 12 12 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 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 12 11 12 12 0 0 0 1 1 0 0 0 0 1 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 12 11 12 12 0 0 1 1 0 0 0 0 0 1 1 0
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 1 1 0 0 0 0 0 12 1 0 0 0 0 0 12 0 0 0 12 0 12

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,11,1,1,0,0,0,12,1,0,0,0,0,12,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,1,11,1,1,0,0,0,12,0,1,0,0,1,12,0,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0,0,1,12,0,0,0,0,0,1,12,12] >;

Dic3×C32⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3^2\rtimes C_4
% in TeX

G:=Group("Dic3xC3^2:C4");
// GroupNames label

G:=SmallGroup(432,567);
// by ID

G=gap.SmallGroup(432,567);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,36,1411,298,1356,1027,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=c^3=d^3=e^4=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
// generators/relations

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