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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, C9○He3⋊4C6, (C3×He3).13C6, He3.15(C3×C6), C33.25(C3×S3), He3.4C6⋊3C3, He3⋊C2.7C32, C32.12(S3×C32), (C3×3- 1+2)⋊18S3, 3- 1+2.3(C3⋊S3), (C3×C9)⋊8(C3×S3), C9.2(C3×C3⋊S3), C32.23(C3×C3⋊S3), C3.13(C32×C3⋊S3), (C3×He3⋊C2).4C3, SmallGroup(486,239)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — He3 — 3- 1+4⋊2C2
Chief series
C1 — C3 — C32 — He3 — C9○He3 — 3- 1+4 — 3- 1+4⋊2C2
Lower central
He3 — 3- 1+4⋊2C2
Upper central
C1 — C3 — 3- 1+2

Generators and relations for 3- 1+4⋊2C2
 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, af=fa, bd=db, fbf=abce-1, ce=ec, fcf=ac-1e-1, de=ed, fdf=d-1e, ef=fe >

Subgroups: 504 in 184 conjugacy classes, 43 normal (10 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, S3×C9, He3⋊C2, C2×3- 1+2, S3×C32, C3×He3, C3×3- 1+2, C3×3- 1+2, C9○He3, C9○He3, S3×3- 1+2, C3×He3⋊C2, He3.4C6, 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 27T167
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 7 4)(2 5 8)(10 16 13)(11 14 17)(19 22 25)(21 27 24)

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

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

(10 26)(11 24)(12 22)(13 20)(14 27)(15 25)(16 23)(17 21)(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), (1,7,4)(2,5,8)(10,16,13)(11,14,17)(19,22,25)(21,27,24), (1,18,19)(2,10,20)(3,11,21)(4,12,22)(5,13,23)(6,14,24)(7,15,25)(8,16,26)(9,17,27), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(20,26,23)(21,24,27), (10,26)(11,24)(12,22)(13,20)(14,27)(15,25)(16,23)(17,21)(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), (1,7,4)(2,5,8)(10,16,13)(11,14,17)(19,22,25)(21,27,24), (1,18,19)(2,10,20)(3,11,21)(4,12,22)(5,13,23)(6,14,24)(7,15,25)(8,16,26)(9,17,27), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(20,26,23)(21,24,27), (10,26)(11,24)(12,22)(13,20)(14,27)(15,25)(16,23)(17,21)(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)], [(1,7,4),(2,5,8),(10,16,13),(11,14,17),(19,22,25),(21,27,24)], [(1,18,19),(2,10,20),(3,11,21),(4,12,22),(5,13,23),(6,14,24),(7,15,25),(8,16,26),(9,17,27)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(20,26,23),(21,24,27)], [(10,26),(11,24),(12,22),(13,20),(14,27),(15,25),(16,23),(17,21),(18,19)]])
 
G:=TransitiveGroup(27,167);
 

58 conjugacy classes

class 1  2 3A3B3C3D3E···3P6A6B6C6D9A···9F9G···9AD18A···18F
order1233333···366669···99···918···18
size1911336···69927273···36···627···27

58 irreducible representations

dim1111112229
type+++
imageC1C2C3C3C6C6S3C3×S3C3×S33- 1+4⋊2C2
kernel3- 1+4⋊2C23- 1+4C3×He3⋊C2He3.4C6C3×He3C9○He3C3×3- 1+2C3×C9C33C1
# reps11262642484

Matrix representation of 3- 1+4⋊2C2 ►in GL9(𝔽19)

700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
0001100000
000010000
000007000
000000100
000000070
0000000011
1100000000
010000000
007000000
,
1100000000
010000000
007000000
000100000
000070000
0000011000
000000700
0000000110
000000001
,
001000000
100000000
010000000
0000011000
0001100000
0000110000
000000007
000000700
000000070
,
100000000
010000000
001000000
000700000
000070000
000007000
0000001100
0000000110
0000000011
,
100000000
001000000
010000000
000100000
000001000
000010000
000000100
000000001
000000010

G:=sub<GL(9,GF(19))| [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],[0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0],[11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[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(486,239);
 
// by ID
 
G=gap.SmallGroup(486,239);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,500,867,3244,382]);
 
// 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,a*f=f*a,b*d=d*b,f*b*f=a*b*c*e^-1,c*e=e*c,f*c*f=a*c^-1*e^-1,d*e=e*d,f*d*f=d^-1*e,e*f=f*e>;
 
// generators/relations
 

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