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## G = A4×He3order 324 = 22·34

### Direct product of A4 and He3

Aliases: A4×He3, C62⋊C32, C32⋊A45C3, C321(C3×A4), C221(C3×He3), (C32×A4)⋊3C3, (C3×A4)⋊1C32, C3.6(C32×A4), (C2×C6).5C33, (C22×He3)⋊5C3, SmallGroup(324,130)

Series: Derived Chief Lower central Upper central

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

Generators and relations for A4×He3
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f3=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=de-1, ef=fe >

Subgroups: 520 in 123 conjugacy classes, 39 normal (8 characteristic)
C1, C2, C3, C3, C22, C6, C32, C32, A4, A4, C2×C6, C2×C6, C3×C6, He3, He3, C33, C3×A4, C3×A4, C3×A4, C62, C2×He3, C3×He3, C32⋊A4, C22×He3, C32×A4, A4×He3
Quotients: C1, C3, C32, A4, He3, C33, C3×A4, C3×He3, C32×A4, A4×He3

Smallest permutation representation of A4×He3
On 36 points
Generators in S36
(1 15)(2 34)(3 19)(4 31)(5 17)(6 27)(7 30)(8 22)(9 12)(10 11)(13 14)(16 18)(20 21)(23 24)(25 26)(28 29)(32 33)(35 36)
(1 13)(2 35)(3 20)(4 32)(5 18)(6 25)(7 28)(8 23)(9 10)(11 12)(14 15)(16 17)(19 21)(22 24)(26 27)(29 30)(31 33)(34 36)
(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 7 6)(2 8 9)(3 5 4)(10 35 23)(11 36 24)(12 34 22)(13 28 25)(14 29 26)(15 30 27)(16 33 21)(17 31 19)(18 32 20)
(1 8 3)(2 4 6)(5 7 9)(10 18 28)(11 16 29)(12 17 30)(13 23 20)(14 24 21)(15 22 19)(25 35 32)(26 36 33)(27 34 31)
(1 5 4)(2 3 9)(6 8 7)(10 35 20)(11 36 21)(12 34 19)(13 18 32)(14 16 33)(15 17 31)(22 30 27)(23 28 25)(24 29 26)

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

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

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

44 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 3K ··· 3P 3Q ··· 3AF 6A 6B 6C ··· 6J order 1 2 3 3 3 ··· 3 3 ··· 3 3 ··· 3 6 6 6 ··· 6 size 1 3 1 1 3 ··· 3 4 ··· 4 12 ··· 12 3 3 9 ··· 9

44 irreducible representations

 dim 1 1 1 1 3 3 3 9 type + + image C1 C3 C3 C3 A4 He3 C3×A4 A4×He3 kernel A4×He3 C32⋊A4 C22×He3 C32×A4 He3 A4 C32 C1 # reps 1 16 2 8 1 6 8 2

Matrix representation of A4×He3 in GL6(𝔽7)

 0 6 1 0 0 0 0 6 0 0 0 0 1 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 6 0 1 0 0 0 6 1 0 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 5 1 0 0 0 0 3 0 0 0 0 0 2 0 2
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 5 2 2 0 0 0 3 0 5 0 0 0 2 0 2

G:=sub<GL(6,GF(7))| [0,0,1,0,0,0,6,6,6,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,6,6,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,5,3,2,0,0,0,1,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,5,3,2,0,0,0,2,0,0,0,0,0,2,5,2] >;

A4×He3 in GAP, Magma, Sage, TeX

A_4\times {\rm He}_3
% in TeX

G:=Group("A4xHe3");
// GroupNames label

G:=SmallGroup(324,130);
// by ID

G=gap.SmallGroup(324,130);
# by ID

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

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

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