He3.2A4 - GroupNames

G = He₃.₂A₄ order 324 = 2²·3⁴

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

Aliases: He₃.₂A₄, C₆².₇C₃², C₃².A₄⋊₇C₃, C₃.₅(C₃²×A₄), C₃².₇(C₃×A₄), (C₂×C₆).₄C₃³, C₂²⋊₂(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

G = He3.2A4 order 324 = 22·34

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

G = He₃.₂A₄ order 324 = 2²·3⁴

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