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## G = He3.A4order 324 = 22·34

### The non-split extension by He3 of A4 acting via A4/C22=C3

Aliases: He3.A4, C62.1C32, (C2×C6).5He3, C32.A42C3, C32.1(C3×A4), C3.6(C32⋊A4), C222(He3.C3), (C22×He3).1C3, (C3×C3.A4)⋊2C3, SmallGroup(324,53)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — He3.A4
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.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=1, f3=b-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 >

Character table of He3.A4

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

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

Matrix representation of He3.A4 in GL6(𝔽19)

 1 6 0 0 0 0 0 18 1 0 0 0 0 18 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 1 7 0 0 0 0 8 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 0 1 0 0 0 0 0 15 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 0 18 0 0 0 0 3 0 18
,
 6 13 10 0 0 0 0 0 10 0 0 0 0 9 10 0 0 0 0 0 0 0 1 0 0 0 0 3 15 17 0 0 0 3 0 4

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,6,18,18,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,1,8,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,15,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,3,0,0,0,0,18,0,0,0,0,0,0,18],[6,0,0,0,0,0,13,0,9,0,0,0,10,10,10,0,0,0,0,0,0,0,3,3,0,0,0,1,15,0,0,0,0,0,17,4] >;

He3.A4 in GAP, Magma, Sage, TeX

{\rm He}_3.A_4
% in TeX

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

G:=SmallGroup(324,53);
// by ID

G=gap.SmallGroup(324,53);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,145,386,4864,8753]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^2=1,f^3=b^-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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