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## G = He3.2A4order 324 = 22·34

### The non-split extension by He3 of A4 acting through Inn(He3)

Aliases: He3.2A4, C62.7C32, C32.A47C3, C3.5(C32×A4), C32.7(C3×A4), (C2×C6).4C33, C222(C9○He3), C3.A4.2C32, (C22×He3).3C3, (C3×C3.A4)⋊7C3, SmallGroup(324,129)

Series: Derived Chief Lower central Upper central

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

Generators and relations for He3.2A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=1, f3=b, ab=ba, cac-1=ab-1, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 205 in 77 conjugacy classes, 36 normal (8 characteristic)
C1, C2, C3, C3, C22, C6, C9, C32, C2×C6, C2×C6, C3×C6, C3×C9, He3, 3- 1+2, C3.A4, C3.A4, C62, C2×He3, C9○He3, C3×C3.A4, C32.A4, C22×He3, He3.2A4
Quotients: C1, C3, C32, A4, C33, C3×A4, C9○He3, C32×A4, He3.2A4

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

44 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 6A 6B 6C ··· 6J 9A ··· 9F 9G ··· 9V order 1 2 3 3 3 ··· 3 6 6 6 ··· 6 9 ··· 9 9 ··· 9 size 1 3 1 1 3 ··· 3 3 3 9 ··· 9 4 ··· 4 12 ··· 12

44 irreducible representations

 dim 1 1 1 1 3 3 3 9 type + + image C1 C3 C3 C3 A4 C3×A4 C9○He3 He3.2A4 kernel He3.2A4 C3×C3.A4 C32.A4 C22×He3 He3 C32 C22 C1 # reps 1 8 16 2 1 8 6 2

Matrix representation of He3.2A4 in GL6(𝔽19)

 7 0 0 0 0 0 0 0 1 0 0 0 10 8 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 7 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
,
 7 1 18 0 0 0 0 0 7 0 0 0 10 8 12 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 1 0 0 0 0 0 18 18 0 0 0 0 8 0 18
,
 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 18 0 0 0 0 11 0 1
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 18 17 0 0 0 0 6 1 1 0 0 0 5 11 0

G:=sub<GL(6,GF(19))| [7,0,10,0,0,0,0,0,8,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,0,10,0,0,0,1,0,8,0,0,0,18,7,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,8,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,11,0,0,0,0,18,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,18,6,5,0,0,0,17,1,11,0,0,0,0,1,0] >;

He3.2A4 in GAP, Magma, Sage, TeX

{\rm He}_3._2A_4
% in TeX

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

G:=SmallGroup(324,129);
// by ID

G=gap.SmallGroup(324,129);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,224,4864,8753]);
// Polycyclic

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

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