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## G = Dic3×He3order 324 = 22·34

### Direct product of Dic3 and He3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — Dic3×He3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×He3 — Dic3×He3
 Lower central C3 — C32 — Dic3×He3
 Upper central C1 — C6 — C2×He3

Generators and relations for Dic3×He3
G = < a,b,c,d,e | a6=c3=d3=e3=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 284 in 100 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C3, C3, C4, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, He3, He3, C33, C3×Dic3, C3×Dic3, C3×C12, C2×He3, C2×He3, C32×C6, C3×He3, C4×He3, C32×Dic3, C6×He3, Dic3×He3
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, He3, C3×Dic3, C3×C12, C2×He3, S3×C32, C4×He3, C32×Dic3, S3×He3, Dic3×He3

Smallest permutation representation of Dic3×He3
On 36 points
Generators in S36
(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)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
(1 17 11)(2 18 12)(3 13 7)(4 14 8)(5 15 9)(6 16 10)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 11 15)(2 12 16)(3 7 17)(4 8 18)(5 9 13)(6 10 14)(19 25 33)(20 26 34)(21 27 35)(22 28 36)(23 29 31)(24 30 32)

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

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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,17,11)(2,18,12)(3,13,7)(4,14,8)(5,15,9)(6,16,10)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,11,15)(2,12,16)(3,7,17)(4,8,18)(5,9,13)(6,10,14)(19,25,33)(20,26,34)(21,27,35)(22,28,36)(23,29,31)(24,30,32) );

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)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)], [(1,17,11),(2,18,12),(3,13,7),(4,14,8),(5,15,9),(6,16,10),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(1,11,15),(2,12,16),(3,7,17),(4,8,18),(5,9,13),(6,10,14),(19,25,33),(20,26,34),(21,27,35),(22,28,36),(23,29,31),(24,30,32)]])

66 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3M 3N ··· 3U 4A 4B 6A 6B 6C 6D 6E 6F ··· 6M 6N ··· 6U 12A 12B 12C 12D 12E ··· 12T order 1 2 3 3 3 3 3 3 ··· 3 3 ··· 3 4 4 6 6 6 6 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 2 3 ··· 3 6 ··· 6 3 3 1 1 2 2 2 3 ··· 3 6 ··· 6 3 3 3 3 9 ··· 9

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 He3 C2×He3 C4×He3 S3×He3 Dic3×He3 kernel Dic3×He3 C6×He3 C32×Dic3 C3×He3 C32×C6 C33 C2×He3 He3 C3×C6 C32 Dic3 C6 C3 C2 C1 # reps 1 1 8 2 8 16 1 1 8 8 2 2 4 2 2

Matrix representation of Dic3×He3 in GL5(𝔽13)

 1 1 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 8 8 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 9 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 1 0 0 0 0 0 3 0

G:=sub<GL(5,GF(13))| [1,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,8,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[9,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0] >;

Dic3×He3 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times {\rm He}_3
% in TeX

G:=Group("Dic3xHe3");
// GroupNames label

G:=SmallGroup(324,93);
// by ID

G=gap.SmallGroup(324,93);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,386,7781]);
// Polycyclic

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

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