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## G = He3⋊2A4order 324 = 22·34

### 2nd semidirect product of He3 and A4 acting via A4/C22=C3

Aliases: He32A4, C62.3C32, C221C3≀C3, (C2×C6).7He3, (C32×A4)⋊1C3, C32.A43C3, C32.3(C3×A4), C3.8(C32⋊A4), (C22×He3)⋊2C3, SmallGroup(324,55)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — He3⋊2A4
 Chief series C1 — C22 — C2×C6 — C62 — C32×A4 — He3⋊2A4
 Lower central C22 — C2×C6 — C62 — He3⋊2A4
 Upper central C1 — C3 — C32 — He3

Generators and relations for He32A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, cac-1=ab-1, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=a-1bc, fdf-1=de=ed, fef-1=d >

3C2
3C3
9C3
12C3
12C3
12C3
3C6
9C6
9C6
9C6
9C6
3C32
12C9
12C32
12C32
12C9
12C32
12C32
3A4
3A4
3A4
4C33
3C62

Character table of He32A4

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 9A 9B 9C 9D size 1 3 1 1 3 3 9 9 12 12 12 12 12 12 3 3 9 9 9 9 9 9 9 9 36 36 36 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ32 1 ζ3 ζ3 ζ3 1 ζ32 ζ3 ζ32 1 1 linear of order 3 ρ3 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 ζ3 ζ3 1 ζ32 ζ32 ζ32 1 ζ3 1 1 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ3 ζ3 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ7 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ32 ζ32 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ8 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ3 1 ζ32 ζ32 ζ32 1 ζ3 ζ32 ζ3 1 1 linear of order 3 ρ9 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 ζ32 ζ32 1 ζ3 ζ3 ζ3 1 ζ32 1 1 ζ32 ζ3 linear of order 3 ρ10 3 -1 3 3 3 3 3 3 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ11 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 3+√-3/2 -3-√-3/2 3-√-3/2 √-3 -3+√-3/2 -√-3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ12 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 3-√-3/2 -3+√-3/2 3+√-3/2 -√-3 -3-√-3/2 √-3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ13 3 -1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 ζ65 ζ65 -1 ζ6 ζ6 ζ6 -1 ζ65 0 0 0 0 complex lifted from C3×A4 ρ14 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -1 -1 -1+√-3 -1-√-3 ζ6 2 -1-√-3 -1+√-3 ζ65 2 0 0 0 0 complex lifted from C32⋊A4 ρ15 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 3 3 0 0 -3-3√-3/2 0 0 0 -3+3√-3/2 0 0 0 0 0 complex lifted from He3 ρ16 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -1 -1 2 -1-√-3 ζ65 -1-√-3 2 -1+√-3 ζ6 -1+√-3 0 0 0 0 complex lifted from C32⋊A4 ρ17 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -1 -1 -1-√-3 2 ζ6 -1-√-3 -1+√-3 2 ζ65 -1+√-3 0 0 0 0 complex lifted from C32⋊A4 ρ18 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 √-3 3+√-3/2 -√-3 -3+√-3/2 3-√-3/2 -3-√-3/2 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ19 3 -1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 ζ6 ζ6 -1 ζ65 ζ65 ζ65 -1 ζ6 0 0 0 0 complex lifted from C3×A4 ρ20 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -1 -1 -1-√-3 -1+√-3 ζ65 2 -1+√-3 -1-√-3 ζ6 2 0 0 0 0 complex lifted from C32⋊A4 ρ21 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -1 -1 -1+√-3 2 ζ65 -1+√-3 -1-√-3 2 ζ6 -1-√-3 0 0 0 0 complex lifted from C32⋊A4 ρ22 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 3 3 0 0 -3+3√-3/2 0 0 0 -3-3√-3/2 0 0 0 0 0 complex lifted from He3 ρ23 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -1 -1 2 -1+√-3 ζ6 -1+√-3 2 -1-√-3 ζ65 -1-√-3 0 0 0 0 complex lifted from C32⋊A4 ρ24 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -3-√-3/2 -√-3 -3+√-3/2 3+√-3/2 √-3 3-√-3/2 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ25 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -3+√-3/2 √-3 -3-√-3/2 3-√-3/2 -√-3 3+√-3/2 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ26 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -√-3 3-√-3/2 √-3 -3-√-3/2 3+√-3/2 -3+√-3/2 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ27 9 -3 -9-9√-3/2 -9+9√-3/2 0 0 0 0 0 0 0 0 0 0 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ28 9 -3 -9+9√-3/2 -9-9√-3/2 0 0 0 0 0 0 0 0 0 0 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of He32A4
On 36 points
Generators in S36
(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 9 5)(2 10 6)(3 11 7)(4 12 8)(13 15 14)(16 17 18)(19 20 21)(22 24 23)(25 26 27)(28 30 29)(31 32 33)(34 36 35)
(1 24 27)(2 35 17)(3 28 20)(4 13 31)(5 22 26)(6 36 16)(7 29 19)(8 14 33)(9 23 25)(10 34 18)(11 30 21)(12 15 32)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 35)(14 36)(15 34)(16 33)(17 31)(18 32)(19 26)(20 27)(21 25)(22 29)(23 30)(24 28)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 28)(14 29)(15 30)(16 26)(17 27)(18 25)(19 33)(20 31)(21 32)(22 36)(23 34)(24 35)
(1 5 9)(2 8 11)(3 6 12)(4 7 10)(13 30 36)(14 28 34)(15 29 35)(16 32 20)(17 33 21)(18 31 19)(22 24 23)(25 27 26)

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

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

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

Matrix representation of He32A4 in GL6(𝔽19)

 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 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 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 1 0 0 0 0 18 0 0 0 0 1 18 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 18 0 1 0 0 0 18 1 0
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,18,18,18,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,18,0,0,0,0,0,1,0,0,0,0,1,0],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

He32A4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_2A_4
% in TeX

G:=Group("He3:2A4");
// GroupNames label

G:=SmallGroup(324,55);
// by ID

G=gap.SmallGroup(324,55);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,224,4864,8753]);
// Polycyclic

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

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