Copied to
clipboard

G = C9⋊C9.S3  order 486 = 2·35

2nd non-split extension by C9⋊C9 of S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C9⋊C9.2S3, C32.He3⋊C2, C3.He3.2C6, C32.3(C32⋊C6), C3.7(He3.2C6), 3- 1+2.S3⋊1C3, (C3×C9).2(C3×S3), SmallGroup(486,39)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3.He3 — C9⋊C9.S3
Chief series
C1 — C3 — C32 — C3×C9 — C3.He3 — C32.He3 — C9⋊C9.S3
Lower central
C3.He3 — C9⋊C9.S3
Upper central
C1

Generators and relations for C9⋊C9.S3
 G = < a,b,c,d | a9=b9=d2=1, c3=a6, bab-1=a7, cac-1=a4b6, dad=a-1, cbc-1=a7b4, bd=db, dcd=a3c2 >

C1
27C2
C3
3C3
54C3
9S3
27C6
C32
3C9
9C9
9C9
18C32
3D9
9D9
9C3×S3
27C18
C3×C9
33- 1+2
3C3×C9
6He3
3C3×D9
9S3×C9
9C9⋊C6
C3.He3
C9⋊C9
2He3⋊C3
3- 1+2.S3
3C9⋊C18
C32.He3
C9⋊C9.S3

Character table of C9⋊C9.S3

 class 123A3B3C3D3E6A6B9A9B9C9D9E9F9G9H18A18B18C18D18E18F
 size 127233545427279999991854272727272727
ρ111111111111111111111111    trivial
ρ21-111111-1-111111111-1-1-1-1-1-1    linear of order 2
ρ31-1111ζ3ζ32-1-1ζ32ζ32ζ3ζ3ζ32ζ311ζ6ζ65ζ65ζ6ζ6ζ65    linear of order 6
ρ411111ζ3ζ3211ζ32ζ32ζ3ζ3ζ32ζ311ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ51-1111ζ32ζ3-1-1ζ3ζ3ζ32ζ32ζ3ζ3211ζ65ζ6ζ6ζ65ζ65ζ6    linear of order 6
ρ611111ζ32ζ311ζ3ζ3ζ32ζ32ζ3ζ3211ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ720222-1-1002222222-1000000    orthogonal lifted from S3
ρ820222ζ65ζ600-1-√-3-1-√-3-1+√-3-1+√-3-1-√-3-1+√-32-1000000    complex lifted from C3×S3
ρ920222ζ6ζ6500-1+√-3-1+√-3-1-√-3-1-√-3-1+√-3-1-√-32-1000000    complex lifted from C3×S3
ρ103-13-3-3√-3/2-3+3√-3/200ζ6ζ65ζ98+2ζ952ζ98+ζ922ζ97+ζ94ζ97+2ζ9ζ95+2ζ922ζ94+ζ900-ζ92-ζ9-ζ94-ζ95-ζ98-ζ97    complex lifted from He3.2C6
ρ11313-3+3√-3/2-3-3√-3/200ζ3ζ322ζ97+ζ942ζ94+ζ92ζ98+ζ92ζ98+2ζ95ζ97+2ζ9ζ95+2ζ9200ζ9ζ95ζ92ζ97ζ94ζ98    complex lifted from He3.2C6
ρ123-13-3-3√-3/2-3+3√-3/200ζ6ζ652ζ98+ζ92ζ95+2ζ922ζ94+ζ92ζ97+ζ94ζ98+2ζ95ζ97+2ζ900-ζ95-ζ97-ζ9-ζ98-ζ92-ζ94    complex lifted from He3.2C6
ρ133-13-3+3√-3/2-3-3√-3/200ζ65ζ6ζ97+2ζ92ζ97+ζ94ζ98+2ζ95ζ95+2ζ922ζ94+ζ92ζ98+ζ9200-ζ94-ζ92-ζ98-ζ9-ζ97-ζ95    complex lifted from He3.2C6
ρ143-13-3+3√-3/2-3-3√-3/200ζ65ζ62ζ97+ζ942ζ94+ζ92ζ98+ζ92ζ98+2ζ95ζ97+2ζ9ζ95+2ζ9200-ζ9-ζ95-ζ92-ζ97-ζ94-ζ98    complex lifted from He3.2C6
ρ15313-3+3√-3/2-3-3√-3/200ζ3ζ32ζ97+2ζ92ζ97+ζ94ζ98+2ζ95ζ95+2ζ922ζ94+ζ92ζ98+ζ9200ζ94ζ92ζ98ζ9ζ97ζ95    complex lifted from He3.2C6
ρ16313-3-3√-3/2-3+3√-3/200ζ32ζ3ζ98+2ζ952ζ98+ζ922ζ97+ζ94ζ97+2ζ9ζ95+2ζ922ζ94+ζ900ζ92ζ9ζ94ζ95ζ98ζ97    complex lifted from He3.2C6
ρ173-13-3-3√-3/2-3+3√-3/200ζ6ζ65ζ95+2ζ92ζ98+2ζ95ζ97+2ζ92ζ94+ζ92ζ98+ζ922ζ97+ζ9400-ζ98-ζ94-ζ97-ζ92-ζ95-ζ9    complex lifted from He3.2C6
ρ18313-3+3√-3/2-3-3√-3/200ζ3ζ322ζ94+ζ9ζ97+2ζ9ζ95+2ζ922ζ98+ζ922ζ97+ζ94ζ98+2ζ9500ζ97ζ98ζ95ζ94ζ9ζ92    complex lifted from He3.2C6
ρ19313-3-3√-3/2-3+3√-3/200ζ32ζ3ζ95+2ζ92ζ98+2ζ95ζ97+2ζ92ζ94+ζ92ζ98+ζ922ζ97+ζ9400ζ98ζ94ζ97ζ92ζ95ζ9    complex lifted from He3.2C6
ρ20313-3-3√-3/2-3+3√-3/200ζ32ζ32ζ98+ζ92ζ95+2ζ922ζ94+ζ92ζ97+ζ94ζ98+2ζ95ζ97+2ζ900ζ95ζ97ζ9ζ98ζ92ζ94    complex lifted from He3.2C6
ρ213-13-3+3√-3/2-3-3√-3/200ζ65ζ62ζ94+ζ9ζ97+2ζ9ζ95+2ζ922ζ98+ζ922ζ97+ζ94ζ98+2ζ9500-ζ97-ζ98-ζ95-ζ94-ζ9-ζ92    complex lifted from He3.2C6
ρ22606660000000000-30000000    orthogonal lifted from C32⋊C6
ρ23180-900000000000000000000    orthogonal faithful

Permutation representations of C9⋊C9.S3
►On 27 points - transitive group 27T152
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)

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

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

(2 9)(3 8)(4 7)(5 6)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)
 
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), (2,5,8)(3,9,6)(10,14,12,13,17,15,16,11,18)(19,27,23,25,24,20,22,21,26), (1,17,26,7,14,23,4,11,20)(2,15,24,8,12,21,5,18,27)(3,13,22,9,10,19,6,16,25), (2,9)(3,8)(4,7)(5,6)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
 
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), (2,5,8)(3,9,6)(10,14,12,13,17,15,16,11,18)(19,27,23,25,24,20,22,21,26), (1,17,26,7,14,23,4,11,20)(2,15,24,8,12,21,5,18,27)(3,13,22,9,10,19,6,16,25), (2,9)(3,8)(4,7)(5,6)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
 
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)], [(2,5,8),(3,9,6),(10,14,12,13,17,15,16,11,18),(19,27,23,25,24,20,22,21,26)], [(1,17,26,7,14,23,4,11,20),(2,15,24,8,12,21,5,18,27),(3,13,22,9,10,19,6,16,25)], [(2,9),(3,8),(4,7),(5,6),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
 
G:=TransitiveGroup(27,152);
 

Matrix representation of C9⋊C9.S3 ►in GL18(ℤ)

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

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

C9⋊C9.S3 in GAP, Magma, Sage, TeX 

C_9\rtimes C_9.S_3
 
% in TeX
 
G:=Group("C9:C9.S3");
 
// GroupNames label
 
G:=SmallGroup(486,39);
 
// by ID
 
G=gap.SmallGroup(486,39);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,3134,224,986,6051,2169,951,453,1096,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^9=b^9=d^2=1,c^3=a^6,b*a*b^-1=a^7,c*a*c^-1=a^4*b^6,d*a*d=a^-1,c*b*c^-1=a^7*b^4,b*d=d*b,d*c*d=a^3*c^2>;
 
// generators/relations
 

Export

Subgroup lattice of C9⋊C9.S3 in TeX
Character table of C9⋊C9.S3 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁