Copied to
clipboard

G = He3.C32  order 243 = 35

4th non-split extension by He3 of C32 acting via C32/C3=C3

p-group, metabelian, nilpotent (class 3), monomial

Aliases: He3.4C32, C32.14He3, C32.7C33, C33.13C32, 3- 1+2⋊4C32, (C3×C9)⋊2C32, He3.C3⋊1C3, (C3×He3).7C3, C3.13(C3×He3), (C3×3- 1+2)⋊8C3, SmallGroup(243,57)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C32 — He3.C32
Chief series
C1 — C3 — C32 — C33 — C3×3- 1+2 — He3.C32
Lower central
C1 — C3 — C32 — He3.C32
Upper central
C1 — C3 — C33 — He3.C32
Jennings
C1 — C3 — C32 — He3.C32

Generators and relations for He3.C32
 G = < a,b,c,d,e | a3=b3=c3=e3=1, d3=b, ab=ba, cac-1=ab-1, ad=da, ae=ea, bc=cb, ede-1=bd=db, be=eb, dcd-1=ab-1c, ce=ec >

Subgroups: 180 in 68 conjugacy classes, 33 normal (8 characteristic)
C1, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, He3.C3, C3×He3, C3×3- 1+2, C3×3- 1+2, He3.C32
Quotients: C1, C3, C32, He3, C33, C3×He3, He3.C32

Permutation representations of He3.C32
►On 27 points - transitive group 27T99
Generators in S27
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)

(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)

(1 20 17)(2 27 18)(3 25 10)(4 23 11)(5 21 12)(6 19 13)(7 26 14)(8 24 15)(9 22 16)

(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)

(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)
 
G:=sub<Sym(27)| (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (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), (1,20,17)(2,27,18)(3,25,10)(4,23,11)(5,21,12)(6,19,13)(7,26,14)(8,24,15)(9,22,16), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)>;
 
G:=Group( (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (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), (1,20,17)(2,27,18)(3,25,10)(4,23,11)(5,21,12)(6,19,13)(7,26,14)(8,24,15)(9,22,16), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24) );
 
G=PermutationGroup([[(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)], [(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)], [(1,20,17),(2,27,18),(3,25,10),(4,23,11),(5,21,12),(6,19,13),(7,26,14),(8,24,15),(9,22,16)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,22,25),(21,27,24)]])
 
G:=TransitiveGroup(27,99);
 

►On 27 points - transitive group 27T109
Generators in S27
(1 20 15)(2 21 16)(3 22 17)(4 23 18)(5 24 10)(6 25 11)(7 26 12)(8 27 13)(9 19 14)

(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)

(2 21 10)(3 17 19)(5 24 13)(6 11 22)(8 27 16)(9 14 25)(12 18 15)(20 23 26)

(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)

(1 4 7)(3 9 6)(11 17 14)(12 15 18)(19 25 22)(20 23 26)
 
G:=sub<Sym(27)| (1,20,15)(2,21,16)(3,22,17)(4,23,18)(5,24,10)(6,25,11)(7,26,12)(8,27,13)(9,19,14), (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), (2,21,10)(3,17,19)(5,24,13)(6,11,22)(8,27,16)(9,14,25)(12,18,15)(20,23,26), (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), (1,4,7)(3,9,6)(11,17,14)(12,15,18)(19,25,22)(20,23,26)>;
 
G:=Group( (1,20,15)(2,21,16)(3,22,17)(4,23,18)(5,24,10)(6,25,11)(7,26,12)(8,27,13)(9,19,14), (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), (2,21,10)(3,17,19)(5,24,13)(6,11,22)(8,27,16)(9,14,25)(12,18,15)(20,23,26), (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), (1,4,7)(3,9,6)(11,17,14)(12,15,18)(19,25,22)(20,23,26) );
 
G=PermutationGroup([[(1,20,15),(2,21,16),(3,22,17),(4,23,18),(5,24,10),(6,25,11),(7,26,12),(8,27,13),(9,19,14)], [(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)], [(2,21,10),(3,17,19),(5,24,13),(6,11,22),(8,27,16),(9,14,25),(12,18,15),(20,23,26)], [(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)], [(1,4,7),(3,9,6),(11,17,14),(12,15,18),(19,25,22),(20,23,26)]])
 
G:=TransitiveGroup(27,109);
 

He3.C32 is a maximal subgroup of   He3.C3⋊C6  C9⋊S3⋊C32  He3.C3⋊2C6

35 conjugacy classes

class 1 3A3B3C···3J3K···3P9A···9R
order1333···33···39···9
size1113···39···99···9

35 irreducible representations

dim111139
type+
imageC1C3C3C3He3He3.C32
kernelHe3.C32He3.C3C3×He3C3×3- 1+2C32C1
# reps1182662

Matrix representation of He3.C32 ►in GL9(𝔽19)

010000000
1286000000
0011000000
000010000
0001286000
0000011000
11120180760
00700110121
00180012080
,
700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
100000000
0110000000
1217000000
0008184000
000100000
0008711000
718011120190
41201671218187
100611111120
,
000100000
000010000
000001000
000000100
11120180760
0000000111
700000000
12100000110
1870000120
,
1100000000
0110000000
0011000000
000100000
000010000
000001000
000000700
18110180070
11120870007

G:=sub<GL(9,GF(19))| [0,12,0,0,0,0,11,0,0,1,8,0,0,0,0,12,0,0,0,6,11,0,0,0,0,7,18,0,0,0,0,12,0,1,0,0,0,0,0,1,8,0,8,0,0,0,0,0,0,6,11,0,11,12,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,6,12,8,0,0,0,0,0,0,0,1,0],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[1,0,12,0,0,0,7,4,1,0,11,1,0,0,0,18,12,0,0,0,7,0,0,0,0,0,0,0,0,0,8,1,8,11,16,6,0,0,0,18,0,7,12,7,11,0,0,0,4,0,11,0,12,1,0,0,0,0,0,0,1,18,11,0,0,0,0,0,0,9,18,12,0,0,0,0,0,0,0,7,0],[0,0,0,0,11,0,7,12,1,0,0,0,0,12,0,0,1,8,0,0,0,0,0,0,0,0,7,1,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,6,11,0,11,12,0,0,0,0,0,1,0,0,0],[11,0,0,0,0,0,0,18,11,0,11,0,0,0,0,0,11,12,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,1,8,0,0,0,0,1,0,0,8,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7] >;
 

He3.C32 in GAP, Magma, Sage, TeX 

{\rm He}_3.C_3^2
 
% in TeX
 
G:=Group("He3.C3^2");
 
// GroupNames label
 
G:=SmallGroup(243,57);
 
// by ID
 
G=gap.SmallGroup(243,57);
 
# by ID
 
G:=PCGroup([5,-3,3,3,-3,-3,301,276,457,2163]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^3=1,d^3=b,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*e=e*a,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,d*c*d^-1=a*b^-1*c,c*e=e*c>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁