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## G = He3.3C12order 324 = 22·34

### The non-split extension by He3 of C12 acting via C12/C3=C4

Aliases: He3.3C12, C9○He3⋊C4, He3⋊C4.C3, C9.(C32⋊C4), He3⋊C2.3C6, He3.4C6.C2, C3.3(C3×C32⋊C4), SmallGroup(324,111)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3.3C12
 Chief series C1 — C3 — He3 — He3⋊C2 — He3.4C6 — He3.3C12
 Lower central He3 — He3.3C12
 Upper central C1 — C9

Generators and relations for He3.3C12
G = < a,b,c,d | a3=b3=c3=1, d12=b, ab=ba, cac-1=ab-1, dad-1=abc, bc=cb, bd=db, dcd-1=ac-1 >

Smallest permutation representation of He3.3C12
On 54 points
Generators in S54
(1 51 21)(2 40 46)(3 35 41)(4 24 30)(5 19 25)(6 44 50)(7 39 45)(8 28 34)(9 23 29)(10 48 54)(11 43 49)(12 32 38)(13 27 33)(14 52 22)(15 47 53)(16 36 42)(17 31 37)(18 20 26)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 43)(20 32 44)(21 33 45)(22 34 46)(23 35 47)(24 36 48)(25 37 49)(26 38 50)(27 39 51)(28 40 52)(29 41 53)(30 42 54)
(2 40 22)(4 24 42)(6 44 26)(8 28 46)(10 48 30)(12 32 50)(14 52 34)(16 36 54)(18 20 38)(19 31 43)(21 45 33)(23 35 47)(25 49 37)(27 39 51)(29 53 41)
(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 49 50 51 52 53 54)

G:=sub<Sym(54)| (1,51,21)(2,40,46)(3,35,41)(4,24,30)(5,19,25)(6,44,50)(7,39,45)(8,28,34)(9,23,29)(10,48,54)(11,43,49)(12,32,38)(13,27,33)(14,52,22)(15,47,53)(16,36,42)(17,31,37)(18,20,26), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,43)(20,32,44)(21,33,45)(22,34,46)(23,35,47)(24,36,48)(25,37,49)(26,38,50)(27,39,51)(28,40,52)(29,41,53)(30,42,54), (2,40,22)(4,24,42)(6,44,26)(8,28,46)(10,48,30)(12,32,50)(14,52,34)(16,36,54)(18,20,38)(19,31,43)(21,45,33)(23,35,47)(25,49,37)(27,39,51)(29,53,41), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)>;

G:=Group( (1,51,21)(2,40,46)(3,35,41)(4,24,30)(5,19,25)(6,44,50)(7,39,45)(8,28,34)(9,23,29)(10,48,54)(11,43,49)(12,32,38)(13,27,33)(14,52,22)(15,47,53)(16,36,42)(17,31,37)(18,20,26), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,43)(20,32,44)(21,33,45)(22,34,46)(23,35,47)(24,36,48)(25,37,49)(26,38,50)(27,39,51)(28,40,52)(29,41,53)(30,42,54), (2,40,22)(4,24,42)(6,44,26)(8,28,46)(10,48,30)(12,32,50)(14,52,34)(16,36,54)(18,20,38)(19,31,43)(21,45,33)(23,35,47)(25,49,37)(27,39,51)(29,53,41), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54) );

G=PermutationGroup([[(1,51,21),(2,40,46),(3,35,41),(4,24,30),(5,19,25),(6,44,50),(7,39,45),(8,28,34),(9,23,29),(10,48,54),(11,43,49),(12,32,38),(13,27,33),(14,52,22),(15,47,53),(16,36,42),(17,31,37),(18,20,26)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,31,43),(20,32,44),(21,33,45),(22,34,46),(23,35,47),(24,36,48),(25,37,49),(26,38,50),(27,39,51),(28,40,52),(29,41,53),(30,42,54)], [(2,40,22),(4,24,42),(6,44,26),(8,28,46),(10,48,30),(12,32,50),(14,52,34),(16,36,54),(18,20,38),(19,31,43),(21,45,33),(23,35,47),(25,49,37),(27,39,51),(29,53,41)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)]])

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 9A ··· 9F 9G 9H 9I 9J 12A 12B 12C 12D 18A ··· 18F 36A ··· 36L order 1 2 3 3 3 3 4 4 6 6 9 ··· 9 9 9 9 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 9 1 1 12 12 9 9 9 9 1 ··· 1 12 12 12 12 9 9 9 9 9 ··· 9 9 ··· 9

42 irreducible representations

 dim 1 1 1 1 1 1 3 4 4 type + + + image C1 C2 C3 C4 C6 C12 He3.3C12 C32⋊C4 C3×C32⋊C4 kernel He3.3C12 He3.4C6 He3⋊C4 C9○He3 He3⋊C2 He3 C1 C9 C3 # reps 1 1 2 2 2 4 24 2 4

Matrix representation of He3.3C12 in GL3(𝔽37) generated by

 0 1 0 0 0 1 1 0 0
,
 10 0 0 0 10 0 0 0 10
,
 1 0 0 0 26 0 0 0 10
,
 17 17 22 17 22 17 35 22 22
G:=sub<GL(3,GF(37))| [0,0,1,1,0,0,0,1,0],[10,0,0,0,10,0,0,0,10],[1,0,0,0,26,0,0,0,10],[17,17,35,17,22,22,22,17,22] >;

He3.3C12 in GAP, Magma, Sage, TeX

{\rm He}_3._3C_{12}
% in TeX

G:=Group("He3.3C12");
// GroupNames label

G:=SmallGroup(324,111);
// by ID

G=gap.SmallGroup(324,111);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,3,-3,36,655,2019,111,2884,916,382]);
// Polycyclic

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

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