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## G = He3⋊C12order 324 = 22·34

### The semidirect product of He3 and C12 acting via C12/C2=C6

Aliases: He3⋊C12, C331Dic3, C2.C3≀S3, C3≀C32C4, He33C41C3, (C2×He3).1C6, (C32×C6).2S3, C3.6(C32⋊C12), C6.11(C32⋊C6), C32.1(C3×Dic3), (C3×C6).1(C3×S3), (C2×C3≀C3).2C2, SmallGroup(324,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C12
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C3≀C3 — He3⋊C12
 Lower central He3 — He3⋊C12
 Upper central C1 — C6

Generators and relations for He3⋊C12
G = < a,b,c,d | a3=b3=c3=d12=1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=ac-1 >

Smallest permutation representation of He3⋊C12
On 36 points
Generators in S36
(2 10 6)(4 12 8)(14 22 18)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)
(1 28 20)(2 21 33)(3 26 22)(4 23 31)(5 36 24)(6 13 29)(7 34 14)(8 15 27)(9 32 16)(10 17 25)(11 30 18)(12 19 35)
(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)

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

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

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

44 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 3I 3J 4A 4B 6A 6B 6C ··· 6H 6I 6J 9A 9B 12A ··· 12P 18A 18B order 1 2 3 3 3 ··· 3 3 3 4 4 6 6 6 ··· 6 6 6 9 9 12 ··· 12 18 18 size 1 1 1 1 3 ··· 3 6 18 9 9 1 1 3 ··· 3 6 18 18 18 9 ··· 9 18 18

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 6 6 type + + + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 C3≀S3 He3⋊C12 C32⋊C6 C32⋊C12 kernel He3⋊C12 C2×C3≀C3 He3⋊3C4 C3≀C3 C2×He3 He3 C32×C6 C33 C3×C6 C32 C2 C1 C6 C3 # reps 1 1 2 2 2 4 1 1 2 2 12 12 1 1

Matrix representation of He3⋊C12 in GL3(𝔽13) generated by

 3 0 0 0 1 0 0 0 9
,
 9 0 0 0 9 0 0 0 9
,
 0 4 0 0 0 7 7 0 0
,
 11 0 0 0 0 4 0 3 0
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,9],[9,0,0,0,9,0,0,0,9],[0,0,7,4,0,0,0,7,0],[11,0,0,0,0,3,0,4,0] >;

He3⋊C12 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_{12}
% in TeX

G:=Group("He3:C12");
// GroupNames label

G:=SmallGroup(324,13);
// by ID

G=gap.SmallGroup(324,13);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,579,585,5404,382]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^12=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations

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