Copied to
clipboard

G = 3+ 1+4⋊2C2  order 486 = 2·35

2nd semidirect product of 3+ 1+4 and C2 acting faithfully

non-abelian, supersoluble, monomial

Aliases: 3+ 1+4⋊2C2, C33⋊6(C3×S3), (C3×He3)⋊12C6, (C3×He3)⋊18S3, He3.7(C3⋊S3), He3.14(C3×C6), He3⋊C2⋊4C32, C32.11(S3×C32), (C3×He3⋊C2)⋊5C3, C32.22(C3×C3⋊S3), C3.12(C32×C3⋊S3), SmallGroup(486,237)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — He3 — 3+ 1+4⋊2C2
Chief series
C1 — C3 — C32 — He3 — C3×He3 — 3+ 1+4 — 3+ 1+4⋊2C2
Lower central
He3 — 3+ 1+4⋊2C2
Upper central
C1 — C3 — He3

Generators and relations for 3+ 1+4⋊2C2
 G = < a,b,c,d,e,f,g | a3=b3=c3=d3=f3=g2=1, e1=b-1, ab=ba, cac-1=ab-1, ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gdg=d-1, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 1098 in 238 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C3×S3, C3×C6, He3, He3, C33, C33, He3⋊C2, C2×He3, S3×C32, C3×He3, C3×He3, S3×He3, C3×He3⋊C2, 3+ 1+4, 3+ 1+4⋊2C2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, S3×C32, C3×C3⋊S3, C32×C3⋊S3, 3+ 1+4⋊2C2

Permutation representations of 3+ 1+4⋊2C2
►On 27 points - transitive group 27T169
Generators in S27
(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 8 26)(2 9 27)(3 7 25)(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)

(2 9 27)(3 25 7)(4 23 19)(5 20 24)(10 15 18)(11 16 13)

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

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

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

(4 15)(5 13)(6 14)(10 19)(11 20)(12 21)(16 24)(17 22)(18 23)
 
G:=sub<Sym(27)| (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), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (2,9,27)(3,25,7)(4,23,19)(5,20,24)(10,15,18)(11,16,13), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,27,19)(11,25,20)(12,26,21), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,12,21)(2,10,19)(3,11,20)(4,9,15)(5,7,13)(6,8,14)(16,24,25)(17,22,26)(18,23,27), (4,15)(5,13)(6,14)(10,19)(11,20)(12,21)(16,24)(17,22)(18,23)>;
 
G:=Group( (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), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (2,9,27)(3,25,7)(4,23,19)(5,20,24)(10,15,18)(11,16,13), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,27,19)(11,25,20)(12,26,21), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,12,21)(2,10,19)(3,11,20)(4,9,15)(5,7,13)(6,8,14)(16,24,25)(17,22,26)(18,23,27), (4,15)(5,13)(6,14)(10,19)(11,20)(12,21)(16,24)(17,22)(18,23) );
 
G=PermutationGroup([[(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)], [(1,8,26),(2,9,27),(3,7,25),(4,23,19),(5,24,20),(6,22,21),(10,15,18),(11,13,16),(12,14,17)], [(2,9,27),(3,25,7),(4,23,19),(5,20,24),(10,15,18),(11,16,13)], [(1,6,14),(2,4,15),(3,5,13),(7,24,16),(8,22,17),(9,23,18),(10,27,19),(11,25,20),(12,26,21)], [(1,26,8),(2,27,9),(3,25,7),(4,19,23),(5,20,24),(6,21,22),(10,18,15),(11,16,13),(12,17,14)], [(1,12,21),(2,10,19),(3,11,20),(4,9,15),(5,7,13),(6,8,14),(16,24,25),(17,22,26),(18,23,27)], [(4,15),(5,13),(6,14),(10,19),(11,20),(12,21),(16,24),(17,22),(18,23)]])
 
G:=TransitiveGroup(27,169);
 

58 conjugacy classes

class 1  2 3A3B3C···3J3K···3AT6A6B6C···6J
order12333···33···3666···6
size19113···36···69927···27

58 irreducible representations

dim1111229
type+++
imageC1C2C3C6S3C3×S33+ 1+4⋊2C2
kernel3+ 1+4⋊2C23+ 1+4C3×He3⋊C2C3×He3C3×He3C33C1
# reps11884324

Matrix representation of 3+ 1+4⋊2C2 ►in GL9(𝔽7)

000100000
000010000
000001000
300500300
000000610
000000601
000000200
010000200
001000200
,
200000000
020000000
002000000
000200000
000020000
000002000
000000200
000000020
000000002
,
100000000
010000000
001000000
000400000
000040000
000004000
100600200
100600020
100600002
,
130000000
061000000
060000000
000130000
000061000
000060000
040020010
040020001
040020100
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
000000004
,
206000000
305000000
215000000
000206000
000305000
000215000
001004002
201104400
601304010
,
100000000
001000000
010000000
000100000
000001000
000010000
000000100
000000001
000000010

G:=sub<GL(9,GF(7))| [0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,6,6,2,2,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0],[2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,6,6,6,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,0,3,6,6,0,0,0,4,4,4,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,6,6,2,2,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[2,3,2,0,0,0,0,2,6,0,0,1,0,0,0,0,0,0,6,5,5,0,0,0,1,1,1,0,0,0,2,3,2,0,1,3,0,0,0,0,0,1,0,0,0,0,0,0,6,5,5,4,4,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;
 

3+ 1+4⋊2C2 in GAP, Magma, Sage, TeX 

3_+^{1+4}\rtimes_2C_2
 
% in TeX
 
G:=Group("ES+(3,2):2C2");
 
// GroupNames label
 
G:=SmallGroup(486,237);
 
// by ID
 
G=gap.SmallGroup(486,237);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,867,3244,382]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^3=c^3=d^3=f^3=g^2=1,e^1=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,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*d*g=d^-1,e*f=f*e,e*g=g*e,g*f*g=f^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁