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G = 3- 1+4⋊C2  order 486 = 2·35

1st semidirect product of 3- 1+4 and C2 acting faithfully

metabelian, supersoluble, monomial

Aliases: 3- 1+4⋊1C2, C9.(S3×C32), C9⋊S3⋊4C32, C9○He3⋊1C6, He3.4S3⋊C3, He3.13(C3×S3), (C3×He3).20S3, C33.24(C3×S3), C33.19(C3⋊S3), C33.S3⋊4C3, C32.21(S3×C32), (C3×3- 1+2)⋊5C6, 3- 1+2⋊9(C3×S3), (C3×3- 1+2)⋊12S3, (C3×C9)⋊4(C3×C6), (C3×C9)⋊4(C3×S3), C3.6(C32×C3⋊S3), C32.14(C3×C3⋊S3), SmallGroup(486,238)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — 3- 1+4⋊C2
Chief series
C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — 3- 1+4 — 3- 1+4⋊C2
Lower central
C3×C9 — 3- 1+4⋊C2
Upper central
C1

Generators and relations for 3- 1+4⋊C2
 G = < a,b,c,d,e,f | a3=c3=d3=e3=f2=1, b3=a, cbc-1=ebe-1=ab=ba, dcd-1=ac=ca, ad=da, ae=ea, faf=a-1, bd=db, fbf=b2d, ce=ec, fcf=c-1, de=ed, fdf=a-1d, ef=fe >

Subgroups: 636 in 181 conjugacy classes, 43 normal (13 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, S3×C32, C3×C3⋊S3, C3×He3, C3×3- 1+2, C3×3- 1+2, C3×3- 1+2, C9○He3, C9○He3, C3×C32⋊C6, C3×C9⋊C6, C33.S3, He3.4S3, 3- 1+4, 3- 1+4⋊C2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, S3×C32, C3×C3⋊S3, C32×C3⋊S3, 3- 1+4⋊C2

Permutation representations of 3- 1+4⋊C2
►On 27 points - transitive group 27T144
Generators in S27
(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 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 19 18)(2 26 13)(3 24 17)(4 22 12)(5 20 16)(6 27 11)(7 25 15)(8 23 10)(9 21 14)

(1 7 4)(2 8 5)(3 9 6)(19 22 25)(20 23 26)(21 24 27)

(2 8 5)(3 6 9)(10 16 13)(11 14 17)(20 26 23)(21 24 27)

(2 9)(3 8)(4 7)(5 6)(10 24)(11 20)(12 25)(13 21)(14 26)(15 22)(16 27)(17 23)(18 19)
 
G:=sub<Sym(27)| (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), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,19,18)(2,26,13)(3,24,17)(4,22,12)(5,20,16)(6,27,11)(7,25,15)(8,23,10)(9,21,14), (1,7,4)(2,8,5)(3,9,6)(19,22,25)(20,23,26)(21,24,27), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(20,26,23)(21,24,27), (2,9)(3,8)(4,7)(5,6)(10,24)(11,20)(12,25)(13,21)(14,26)(15,22)(16,27)(17,23)(18,19)>;
 
G:=Group( (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), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,19,18)(2,26,13)(3,24,17)(4,22,12)(5,20,16)(6,27,11)(7,25,15)(8,23,10)(9,21,14), (1,7,4)(2,8,5)(3,9,6)(19,22,25)(20,23,26)(21,24,27), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(20,26,23)(21,24,27), (2,9)(3,8)(4,7)(5,6)(10,24)(11,20)(12,25)(13,21)(14,26)(15,22)(16,27)(17,23)(18,19) );
 
G=PermutationGroup([[(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)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,19,18),(2,26,13),(3,24,17),(4,22,12),(5,20,16),(6,27,11),(7,25,15),(8,23,10),(9,21,14)], [(1,7,4),(2,8,5),(3,9,6),(19,22,25),(20,23,26),(21,24,27)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(20,26,23),(21,24,27)], [(2,9),(3,8),(4,7),(5,6),(10,24),(11,20),(12,25),(13,21),(14,26),(15,22),(16,27),(17,23),(18,19)]])
 
G:=TransitiveGroup(27,144);
 

►On 27 points - transitive group 27T145
Generators in S27
(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 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 16 13)(11 14 17)(19 25 22)(20 23 26)

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

(2 8 5)(3 6 9)(11 17 14)(12 15 18)(19 22 25)(21 27 24)

(2 22)(3 11)(4 7)(5 19)(6 17)(8 25)(9 14)(10 16)(12 24)(15 21)(18 27)(20 26)
 
G:=sub<Sym(27)| (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), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26), (1,20,16)(2,21,17)(3,22,18)(4,23,10)(5,24,11)(6,25,12)(7,26,13)(8,27,14)(9,19,15), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,22,25)(21,27,24), (2,22)(3,11)(4,7)(5,19)(6,17)(8,25)(9,14)(10,16)(12,24)(15,21)(18,27)(20,26)>;
 
G:=Group( (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), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26), (1,20,16)(2,21,17)(3,22,18)(4,23,10)(5,24,11)(6,25,12)(7,26,13)(8,27,14)(9,19,15), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,22,25)(21,27,24), (2,22)(3,11)(4,7)(5,19)(6,17)(8,25)(9,14)(10,16)(12,24)(15,21)(18,27)(20,26) );
 
G=PermutationGroup([[(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)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,25,22),(20,23,26)], [(1,20,16),(2,21,17),(3,22,18),(4,23,10),(5,24,11),(6,25,12),(7,26,13),(8,27,14),(9,19,15)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(19,22,25),(21,27,24)], [(2,22),(3,11),(4,7),(5,19),(6,17),(8,25),(9,14),(10,16),(12,24),(15,21),(18,27),(20,26)]])
 
G:=TransitiveGroup(27,145);
 

55 conjugacy classes

class 1  2 3A3B···3I3J···3R6A···6H9A···9AA
order1233···33···36···69···9
size12723···36···627···276···6

55 irreducible representations

dim11111118222222
type+++++
imageC1C2C3C3C6C63- 1+4⋊C2S3S3C3×S3C3×S3C3×S3C3×S3
kernel3- 1+4⋊C23- 1+4C33.S3He3.4S3C3×3- 1+2C9○He3C1C3×He3C3×3- 1+2C3×C9He33- 1+2C33
# reps11262611366182

Matrix representation of 3- 1+4⋊C2 ►in GL18(ℤ)

-1-10000000000000000
100000000000000000
00-1-100000000000000
001000000000000000
0000-1-1000000000000
000010000000000000
000000-1-10000000000
000000100000000000
00000000-1-100000000
000000001000000000
0000000000-1-1000000
000000000010000000
000000000000-1-10000
000000000000100000
00000000000000-1-100
000000000000001000
0000000000000000-1-1
000000000000000010
,
000000100000000000
000000010000000000
000000000100000000
00000000-1-100000000
0000000000-1-1000000
000000000010000000
000000000000100000
000000000000010000
000000000000000100
00000000000000-1-100
0000000000000000-1-1
000000000000000010
-1-10000000000000000
100000000000000000
001000000000000000
000100000000000000
000001000000000000
0000-1-1000000000000
,
001000000000000000
000100000000000000
000010000000000000
000001000000000000
100000000000000000
010000000000000000
000000001000000000
000000000100000000
000000000010000000
000000000001000000
000000100000000000
000000010000000000
000000000000001000
000000000000000100
000000000000000010
000000000000000001
000000000000100000
000000000000010000
,
010000000000000000
-1-10000000000000000
001000000000000000
000100000000000000
0000-1-1000000000000
000010000000000000
000000010000000000
000000-1-10000000000
000000001000000000
000000000100000000
0000000000-1-1000000
000000000010000000
000000000000010000
000000000000-1-10000
000000000000001000
000000000000000100
0000000000000000-1-1
000000000000000010
,
100000000000000000
010000000000000000
001000000000000000
000100000000000000
000010000000000000
000001000000000000
000000-1-10000000000
000000100000000000
00000000-1-100000000
000000001000000000
0000000000-1-1000000
000000000010000000
000000000000010000
000000000000-1-10000
000000000000000100
00000000000000-1-100
000000000000000001
0000000000000000-1-1
,
100000000000000000
-1-10000000000000000
000010000000000000
0000-1-1000000000000
001000000000000000
00-1-100000000000000
000000000000010000
000000000000100000
000000000000000001
000000000000000010
000000000000000100
000000000000001000
000000010000000000
000000100000000000
000000000001000000
000000000010000000
000000000100000000
000000001000000000

G:=sub<GL(18,Integers())| [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0] >;
 

3- 1+4⋊C2 in GAP, Magma, Sage, TeX 

3_-^{1+4}\rtimes C_2
 
% in TeX
 
G:=Group("ES-(3,2):C2");
 
// GroupNames label
 
G:=SmallGroup(486,238);
 
// by ID
 
G=gap.SmallGroup(486,238);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,735,453,3244,3250,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=c^3=d^3=e^3=f^2=1,b^3=a,c*b*c^-1=e*b*e^-1=a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*d=d*b,f*b*f=b^2*d,c*e=e*c,f*c*f=c^-1,d*e=e*d,f*d*f=a^-1*d,e*f=f*e>;
 
// generators/relations
 

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