Copied to
clipboard

G = C34⋊4C6  order 486 = 2·35

4th semidirect product of C34 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C34⋊4C6, (C3×He3)⋊8S3, C32⋊He3⋊4C2, C33⋊1(C3⋊S3), C34⋊C2⋊1C3, C33.62(C3×S3), C32⋊3(C32⋊C6), C3.3(He3⋊4S3), C32.36(C3×C3⋊S3), SmallGroup(486,146)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C34 — C34⋊4C6
Chief series
C1 — C3 — C32 — C33 — C34 — C32⋊He3 — C34⋊4C6
Lower central
C34 — C34⋊4C6
Upper central
C1

Generators and relations for C34⋊4C6
 G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, eae-1=a-1c-1, bc=cb, bd=db, ebe-1=b-1d-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 3302 in 258 conjugacy classes, 31 normal (7 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, He3, C33, C33, C33, C32⋊C6, C3×C3⋊S3, C33⋊C2, C3×He3, C3×He3, C34, He3⋊4S3, C34⋊C2, C32⋊He3, C34⋊4C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He3⋊4S3, C34⋊4C6

Character table of C34⋊4C6

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q3R3S3T3U3V3W3X3Y3Z6A6B
 size 18122226666666666669918181818181818188181
ρ1111111111111111111111111111111    trivial
ρ21-111111111111111111111111111-1-1    linear of order 2
ρ3111111111111111111ζ3ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ41-11111111111111111ζ3ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ65ζ6    linear of order 6
ρ5111111111111111111ζ32ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ61-11111111111111111ζ32ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ6ζ65    linear of order 6
ρ7202222-1-1222-1-1-1-1-1-1-122-1-12-12-1-1-100    orthogonal lifted from S3
ρ82022222-1-1-1-1-1-1-12-12-122-12-1-1-1-12-100    orthogonal lifted from S3
ρ9202222-12-1-1-1-1-1-1-12-12222-1-12-1-1-1-100    orthogonal lifted from S3
ρ10202222-1-1-1-1-1222-1-1-1-122-1-1-1-1-12-1200    orthogonal lifted from S3
ρ11202222-1-1-1-1-1222-1-1-1-1-1-√-3-1+√-3ζ6ζ6ζ6ζ65ζ65-1+√-3ζ65-1-√-300    complex lifted from C3×S3
ρ12202222-12-1-1-1-1-1-1-12-12-1-√-3-1+√-3-1-√-3ζ6ζ6-1+√-3ζ65ζ65ζ65ζ600    complex lifted from C3×S3
ρ132022222-1-1-1-1-1-1-12-12-1-1-√-3-1+√-3ζ6-1-√-3ζ6ζ65ζ65ζ65-1+√-3ζ600    complex lifted from C3×S3
ρ14202222-1-1222-1-1-1-1-1-1-1-1-√-3-1+√-3ζ6ζ6-1-√-3ζ65-1+√-3ζ65ζ65ζ600    complex lifted from C3×S3
ρ15202222-1-1222-1-1-1-1-1-1-1-1+√-3-1-√-3ζ65ζ65-1+√-3ζ6-1-√-3ζ6ζ6ζ6500    complex lifted from C3×S3
ρ16202222-12-1-1-1-1-1-1-12-12-1+√-3-1-√-3-1+√-3ζ65ζ65-1-√-3ζ6ζ6ζ6ζ6500    complex lifted from C3×S3
ρ17202222-1-1-1-1-1222-1-1-1-1-1+√-3-1-√-3ζ65ζ65ζ65ζ6ζ6-1-√-3ζ6-1+√-300    complex lifted from C3×S3
ρ182022222-1-1-1-1-1-1-12-12-1-1+√-3-1-√-3ζ65-1+√-3ζ65ζ6ζ6ζ6-1-√-3ζ6500    complex lifted from C3×S3
ρ1960-3-3-360-3000000060-3000000000000    orthogonal lifted from C32⋊C6
ρ20606-3-3-300-36-30000000000000000000    orthogonal lifted from C32⋊C6
ρ2160-36-3-300000-3-360000000000000000    orthogonal lifted from C32⋊C6
ρ2260-3-36-3-30000000-3060000000000000    orthogonal lifted from C32⋊C6
ρ2360-3-36-3-3000000060-30000000000000    orthogonal lifted from C32⋊C6
ρ2460-3-3-36060000000-30-3000000000000    orthogonal lifted from C32⋊C6
ρ2560-3-36-360000000-30-30000000000000    orthogonal lifted from C32⋊C6
ρ26606-3-3-300-3-360000000000000000000    orthogonal lifted from C32⋊C6
ρ2760-3-3-360-30000000-306000000000000    orthogonal lifted from C32⋊C6
ρ28606-3-3-3006-3-30000000000000000000    orthogonal lifted from C32⋊C6
ρ2960-36-3-3000006-3-30000000000000000    orthogonal lifted from C32⋊C6
ρ3060-36-3-300000-36-30000000000000000    orthogonal lifted from C32⋊C6

Permutation representations of C34⋊4C6
►On 27 points - transitive group 27T197
Generators in S27
(1 21 18)(2 9 6)(3 14 11)(4 24 17)(5 15 22)(7 20 27)(8 25 12)(10 16 26)(13 23 19)

(1 15 12)(3 14 11)(4 24 17)(5 25 18)(7 20 27)(8 21 22)

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

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

(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)
 
G:=sub<Sym(27)| (1,21,18)(2,9,6)(3,14,11)(4,24,17)(5,15,22)(7,20,27)(8,25,12)(10,16,26)(13,23,19), (1,15,12)(3,14,11)(4,24,17)(5,25,18)(7,20,27)(8,21,22), (1,22,25)(2,26,23)(3,24,27)(4,20,11)(5,12,21)(6,16,13)(7,14,17)(8,18,15)(9,10,19), (1,12,15)(2,10,13)(3,14,11)(4,24,17)(5,18,25)(6,26,19)(7,20,27)(8,22,21)(9,16,23), (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)>;
 
G:=Group( (1,21,18)(2,9,6)(3,14,11)(4,24,17)(5,15,22)(7,20,27)(8,25,12)(10,16,26)(13,23,19), (1,15,12)(3,14,11)(4,24,17)(5,25,18)(7,20,27)(8,21,22), (1,22,25)(2,26,23)(3,24,27)(4,20,11)(5,12,21)(6,16,13)(7,14,17)(8,18,15)(9,10,19), (1,12,15)(2,10,13)(3,14,11)(4,24,17)(5,18,25)(6,26,19)(7,20,27)(8,22,21)(9,16,23), (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) );
 
G=PermutationGroup([[(1,21,18),(2,9,6),(3,14,11),(4,24,17),(5,15,22),(7,20,27),(8,25,12),(10,16,26),(13,23,19)], [(1,15,12),(3,14,11),(4,24,17),(5,25,18),(7,20,27),(8,21,22)], [(1,22,25),(2,26,23),(3,24,27),(4,20,11),(5,12,21),(6,16,13),(7,14,17),(8,18,15),(9,10,19)], [(1,12,15),(2,10,13),(3,14,11),(4,24,17),(5,18,25),(6,26,19),(7,20,27),(8,22,21),(9,16,23)], [(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)]])
 
G:=TransitiveGroup(27,197);
 

Matrix representation of C34⋊4C6 ►in GL12(ℤ)

-110000000000
-100000000000
000-100000000
001-100000000
000010000000
000001000000
000000010000
000000-1-10000
000000001000
000000000100
0000000000-1-1
000000000010
,
-110000000000
-100000000000
000-100000000
001-100000000
000010000000
000001000000
000000010000
000000-1-10000
00000000-1-100
000000001000
000000000010
000000000001
,
-110000000000
-100000000000
00-1100000000
00-1000000000
0000-11000000
0000-10000000
000000-1-10000
000000100000
00000000-1-100
000000001000
0000000000-1-1
000000000010
,
-110000000000
-100000000000
00-1100000000
00-1000000000
0000-11000000
0000-10000000
000000010000
000000-1-10000
000000000100
00000000-1-100
000000000001
0000000000-1-1
,
00001-1000000
00000-1000000
1-10000000000
0-10000000000
001-100000000
000-100000000
000000000010
0000000000-1-1
000000100000
000000-1-10000
000000001000
00000000-1-100

G:=sub<GL(12,Integers())| [-1,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0],[-1,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-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,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,-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,-1,-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,-1,-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,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0],[-1,-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,-1,-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,-1,-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,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,-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,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-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,-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,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0] >;
 

C34⋊4C6 in GAP, Magma, Sage, TeX 

C_3^4\rtimes_4C_6
 
% in TeX
 
G:=Group("C3^4:4C6");
 
// GroupNames label
 
G:=SmallGroup(486,146);
 
// by ID
 
G=gap.SmallGroup(486,146);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,548,867,2169,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1*c^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1*d^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Character table of C34⋊4C6 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁