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G = He3.2A4order 324 = 22·34

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

metabelian, soluble, monomial

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

C1C2×C6 — He3.2A4
C1C22C2×C6C3.A4C3×C3.A4 — He3.2A4
C22C2×C6 — He3.2A4
C1C3He3

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 3A3B3C···3J6A6B6C···6J9A···9F9G···9V
order12333···3666···69···99···9
size13113···3339···94···412···12

44 irreducible representations

dim11113339
type++
imageC1C3C3C3A4C3×A4C9○He3He3.2A4
kernelHe3.2A4C3×C3.A4C32.A4C22×He3He3C32C22C1
# reps181621862

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

700000
001000
10812000
000100
000010
000001
,
700000
070000
007000
000100
000010
000001
,
7118000
007000
10812000
000100
000010
000001
,
100000
010000
001000
000100
00018180
0008018
,
100000
010000
001000
0001800
0000180
0001101
,
900000
090000
009000
00018170
000611
0005110

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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