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## G = He3⋊A4order 324 = 22·34

### 1st semidirect product of He3 and A4 acting via A4/C22=C3

Aliases: He31A4, C62.2C32, C32⋊A42C3, (C2×C6).6He3, C32.2(C3×A4), C3.7(C32⋊A4), (C22×He3)⋊1C3, C222(He3⋊C3), (C3×C3.A4)⋊3C3, SmallGroup(324,54)

Series: Derived Chief Lower central Upper central

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

Generators and relations for He3⋊A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, cac-1=faf-1=ab-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=a-1b-1c, fdf-1=de=ed, fef-1=d >

Character table of He3⋊A4

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 9A 9B 9C 9D 9E 9F size 1 3 1 1 3 3 9 9 36 36 36 36 3 3 9 9 9 9 9 9 9 9 12 12 12 12 12 12 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 ζ3 ζ32 1 ζ3 1 ζ32 1 1 ζ32 ζ3 ζ3 ζ3 1 ζ32 ζ32 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ3 1 1 1 1 1 1 ζ3 ζ32 ζ3 1 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ3 1 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 ζ32 ζ3 ζ3 ζ3 1 ζ32 ζ32 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ7 1 1 1 1 1 1 ζ32 ζ3 ζ32 1 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ32 1 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ8 1 1 1 1 1 1 ζ32 ζ3 1 ζ32 1 ζ3 1 1 ζ3 ζ32 ζ32 ζ32 1 ζ3 ζ3 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ9 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 ζ3 ζ32 ζ32 ζ32 1 ζ3 ζ3 1 1 1 1 1 1 1 linear of order 3 ρ10 3 -1 3 3 3 3 3 3 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ11 3 -1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -1 -1 ζ6 ζ65 ζ65 ζ65 -1 ζ6 ζ6 -1 0 0 0 0 0 0 complex lifted from C3×A4 ρ12 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 -1-√-3 2 -1+√-3 -1-√-3 ζ6 -1+√-3 2 ζ65 0 0 0 0 0 0 complex lifted from C32⋊A4 ρ13 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 2 -1+√-3 2 -1-√-3 ζ65 -1+√-3 -1-√-3 ζ6 0 0 0 0 0 0 complex lifted from C32⋊A4 ρ14 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 3 3 0 0 0 0 -3-3√-3/2 0 0 -3+3√-3/2 0 0 0 0 0 0 complex lifted from He3 ρ15 3 -1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -1 -1 ζ65 ζ6 ζ6 ζ6 -1 ζ65 ζ65 -1 0 0 0 0 0 0 complex lifted from C3×A4 ρ16 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 3 3 0 0 0 0 -3+3√-3/2 0 0 -3-3√-3/2 0 0 0 0 0 0 complex lifted from He3 ρ17 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 -1+√-3 -1+√-3 -1-√-3 2 ζ6 2 -1-√-3 ζ65 0 0 0 0 0 0 complex lifted from C32⋊A4 ρ18 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 -1+√-3 2 -1-√-3 -1+√-3 ζ65 -1-√-3 2 ζ6 0 0 0 0 0 0 complex lifted from C32⋊A4 ρ19 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 2 -1-√-3 2 -1+√-3 ζ6 -1-√-3 -1+√-3 ζ65 0 0 0 0 0 0 complex lifted from C32⋊A4 ρ20 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 -1-√-3 -1-√-3 -1+√-3 2 ζ65 2 -1+√-3 ζ6 0 0 0 0 0 0 complex lifted from C32⋊A4 ρ21 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 ζ98+2ζ95 ζ95+2ζ92 2ζ97+ζ94 2ζ94+ζ9 ζ97+2ζ9 2ζ98+ζ92 complex lifted from He3⋊C3 ρ22 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 ζ97+2ζ9 2ζ94+ζ9 ζ98+2ζ95 2ζ98+ζ92 ζ95+2ζ92 2ζ97+ζ94 complex lifted from He3⋊C3 ρ23 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 2ζ97+ζ94 ζ97+2ζ9 2ζ98+ζ92 ζ95+2ζ92 ζ98+2ζ95 2ζ94+ζ9 complex lifted from He3⋊C3 ρ24 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 2ζ94+ζ9 2ζ97+ζ94 ζ95+2ζ92 ζ98+2ζ95 2ζ98+ζ92 ζ97+2ζ9 complex lifted from He3⋊C3 ρ25 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 ζ95+2ζ92 2ζ98+ζ92 ζ97+2ζ9 2ζ97+ζ94 2ζ94+ζ9 ζ98+2ζ95 complex lifted from He3⋊C3 ρ26 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 2ζ98+ζ92 ζ98+2ζ95 2ζ94+ζ9 ζ97+2ζ9 2ζ97+ζ94 ζ95+2ζ92 complex lifted from He3⋊C3 ρ27 9 -3 -9-9√-3/2 -9+9√-3/2 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 0 0 complex faithful ρ28 9 -3 -9+9√-3/2 -9-9√-3/2 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 0 0 complex faithful

Smallest permutation representation of He3⋊A4
On 54 points
Generators in S54
(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)
(1 10 22)(2 11 23)(3 12 24)(4 16 26)(5 17 27)(6 18 25)(7 43 49)(8 44 50)(9 45 51)(13 53 41)(14 54 42)(15 52 40)(19 36 48)(20 34 46)(21 35 47)(28 38 32)(29 39 33)(30 37 31)
(2 11 23)(3 24 12)(4 26 16)(6 18 25)(7 45 44)(8 49 9)(13 52 54)(14 41 15)(19 34 21)(20 47 48)(28 39 30)(29 31 32)(33 37 38)(35 36 46)(40 42 53)(43 51 50)
(1 17)(2 18)(3 16)(4 24)(5 22)(6 23)(7 53)(8 54)(9 52)(10 27)(11 25)(12 26)(13 49)(14 50)(15 51)(40 45)(41 43)(42 44)
(7 53)(8 54)(9 52)(13 49)(14 50)(15 51)(19 38)(20 39)(21 37)(28 48)(29 46)(30 47)(31 35)(32 36)(33 34)(40 45)(41 43)(42 44)
(1 9 29)(2 43 31)(3 50 38)(4 42 48)(5 15 34)(6 53 21)(7 37 23)(8 32 12)(10 45 39)(11 49 30)(13 47 25)(14 19 16)(17 52 46)(18 41 35)(20 27 40)(22 51 33)(24 44 28)(26 54 36)

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

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

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

Matrix representation of He3⋊A4 in GL6(𝔽19)

 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
,
 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
,
 1 0 0 0 0 0 0 7 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
,
 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 14 1 0 0 0 0 0 0 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 5 18 0 0 0 0 9 0 18
,
 13 15 15 0 0 0 10 15 10 0 0 0 13 13 10 0 0 0 0 0 0 5 17 0 0 0 0 8 14 1 0 0 0 3 10 0

G:=sub<GL(6,GF(19))| [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],[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],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,14,0,0,0,0,0,1,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,9,0,0,0,0,18,0,0,0,0,0,0,18],[13,10,13,0,0,0,15,15,13,0,0,0,15,10,10,0,0,0,0,0,0,5,8,3,0,0,0,17,14,10,0,0,0,0,1,0] >;

He3⋊A4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(324,54);
// by ID

G=gap.SmallGroup(324,54);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,650,224,4864,8753]);
// 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=f*a*f^-1=a*b^-1,a*d=d*a,a*e=e*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^-1*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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