Copied to
clipboard

G = (C3×He3)⋊C6  order 486 = 2·35

8th semidirect product of C3×He3 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C3≀C3⋊3C6, C33⋊C6⋊C3, (C3×He3)⋊8C6, (C3×He3)⋊6S3, C33⋊2(C3×C6), He3.3(C3×S3), He3⋊4S3⋊2C3, C33.14(C3×S3), C33⋊C2⋊2C32, C33⋊C32⋊3C2, C32.7(C32⋊C6), C32.16(S3×C32), C3.7(C3×C32⋊C6), SmallGroup(486,127)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — (C3×He3)⋊C6
Chief series
C1 — C3 — C32 — C33 — C3×He3 — C33⋊C32 — (C3×He3)⋊C6
Lower central
C33 — (C3×He3)⋊C6
Upper central
C1

Generators and relations for (C3×He3)⋊C6
 G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, ab=ba, ac=ca, ede-1=ad=da, eae-1=a-1c-1, bc=cb, dbd-1=bc-1, ebe-1=a-1b-1, cd=dc, ece-1=c-1 >

Subgroups: 654 in 82 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C3×S3, C3⋊S3, C3×C6, C3×C9, He3, He3, 3- 1+2, C33, C33, C32⋊C6, S3×C32, C3×C3⋊S3, C33⋊C2, C3≀C3, C3≀C3, C3×He3, C3×3- 1+2, C33⋊C6, C3×C32⋊C6, He3⋊4S3, C33⋊C32, (C3×He3)⋊C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, (C3×He3)⋊C6

Permutation representations of (C3×He3)⋊C6
►On 27 points - transitive group 27T155
Generators in S27
(1 13 10)(3 12 15)(4 23 26)(5 24 27)(8 21 18)(9 16 19)

(2 11 14)(3 15 12)(5 27 24)(7 17 20)(8 21 18)(9 16 19)

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

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

(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)| (1,13,10)(3,12,15)(4,23,26)(5,24,27)(8,21,18)(9,16,19), (2,11,14)(3,15,12)(5,27,24)(7,17,20)(8,21,18)(9,16,19), (1,10,13)(2,14,11)(3,12,15)(4,23,26)(5,27,24)(6,25,22)(7,17,20)(8,21,18)(9,19,16), (1,9,5)(2,17,22)(3,18,23)(4,15,21)(6,14,20)(7,25,11)(8,26,12)(10,19,27)(13,16,24), (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( (1,13,10)(3,12,15)(4,23,26)(5,24,27)(8,21,18)(9,16,19), (2,11,14)(3,15,12)(5,27,24)(7,17,20)(8,21,18)(9,16,19), (1,10,13)(2,14,11)(3,12,15)(4,23,26)(5,27,24)(6,25,22)(7,17,20)(8,21,18)(9,19,16), (1,9,5)(2,17,22)(3,18,23)(4,15,21)(6,14,20)(7,25,11)(8,26,12)(10,19,27)(13,16,24), (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([[(1,13,10),(3,12,15),(4,23,26),(5,24,27),(8,21,18),(9,16,19)], [(2,11,14),(3,15,12),(5,27,24),(7,17,20),(8,21,18),(9,16,19)], [(1,10,13),(2,14,11),(3,12,15),(4,23,26),(5,27,24),(6,25,22),(7,17,20),(8,21,18),(9,19,16)], [(1,9,5),(2,17,22),(3,18,23),(4,15,21),(6,14,20),(7,25,11),(8,26,12),(10,19,27),(13,16,24)], [(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(27,155);
 

►On 27 points - transitive group 27T171
Generators in S27
(1 23 26)(2 24 27)(4 17 10)(5 18 11)(7 13 20)(8 14 21)

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

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

(1 26 23)(3 25 22)(5 18 11)(6 12 19)(7 13 20)(8 21 14)

(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)| (1,23,26)(2,24,27)(4,17,10)(5,18,11)(7,13,20)(8,14,21), (1,10,13)(2,8,5)(3,12,15)(4,20,23)(6,16,25)(7,26,17)(9,22,19)(11,27,21)(14,18,24), (1,23,26)(2,27,24)(3,25,22)(4,17,10)(5,11,18)(6,19,12)(7,13,20)(8,21,14)(9,15,16), (1,26,23)(3,25,22)(5,18,11)(6,12,19)(7,13,20)(8,21,14), (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( (1,23,26)(2,24,27)(4,17,10)(5,18,11)(7,13,20)(8,14,21), (1,10,13)(2,8,5)(3,12,15)(4,20,23)(6,16,25)(7,26,17)(9,22,19)(11,27,21)(14,18,24), (1,23,26)(2,27,24)(3,25,22)(4,17,10)(5,11,18)(6,19,12)(7,13,20)(8,21,14)(9,15,16), (1,26,23)(3,25,22)(5,18,11)(6,12,19)(7,13,20)(8,21,14), (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([[(1,23,26),(2,24,27),(4,17,10),(5,18,11),(7,13,20),(8,14,21)], [(1,10,13),(2,8,5),(3,12,15),(4,20,23),(6,16,25),(7,26,17),(9,22,19),(11,27,21),(14,18,24)], [(1,23,26),(2,27,24),(3,25,22),(4,17,10),(5,11,18),(6,19,12),(7,13,20),(8,21,14),(9,15,16)], [(1,26,23),(3,25,22),(5,18,11),(6,12,19),(7,13,20),(8,21,14)], [(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(27,171);
 

►On 27 points - transitive group 27T182
Generators in S27
(1 5 8)(2 9 4)(3 7 6)(10 25 12)(11 19 21)(13 15 22)(14 18 16)(17 24 26)(20 23 27)

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

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

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

(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)| (1,5,8)(2,9,4)(3,7,6)(10,25,12)(11,19,21)(13,15,22)(14,18,16)(17,24,26)(20,23,27), (1,13,10)(2,24,16)(3,19,27)(4,17,18)(5,15,25)(6,11,23)(7,21,20)(8,22,12)(9,26,14), (1,3,2)(4,8,6)(5,7,9)(10,27,16)(11,17,22)(12,23,18)(13,19,24)(14,25,20)(15,21,26), (1,7,4)(2,5,6)(3,9,8)(10,25,12)(11,13,26)(14,18,16)(15,17,19)(20,23,27)(21,22,24), (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( (1,5,8)(2,9,4)(3,7,6)(10,25,12)(11,19,21)(13,15,22)(14,18,16)(17,24,26)(20,23,27), (1,13,10)(2,24,16)(3,19,27)(4,17,18)(5,15,25)(6,11,23)(7,21,20)(8,22,12)(9,26,14), (1,3,2)(4,8,6)(5,7,9)(10,27,16)(11,17,22)(12,23,18)(13,19,24)(14,25,20)(15,21,26), (1,7,4)(2,5,6)(3,9,8)(10,25,12)(11,13,26)(14,18,16)(15,17,19)(20,23,27)(21,22,24), (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([[(1,5,8),(2,9,4),(3,7,6),(10,25,12),(11,19,21),(13,15,22),(14,18,16),(17,24,26),(20,23,27)], [(1,13,10),(2,24,16),(3,19,27),(4,17,18),(5,15,25),(6,11,23),(7,21,20),(8,22,12),(9,26,14)], [(1,3,2),(4,8,6),(5,7,9),(10,27,16),(11,17,22),(12,23,18),(13,19,24),(14,25,20),(15,21,26)], [(1,7,4),(2,5,6),(3,9,8),(10,25,12),(11,13,26),(14,18,16),(15,17,19),(20,23,27),(21,22,24)], [(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(27,182);
 

31 conjugacy classes

class 1  2 3A3B3C3D3E3F3G···3L3M3N3O6A···6H9A···9F
order123333333···33336···69···9
size1272336669···918181827···2718···18

31 irreducible representations

dim1111111822266
type+++++
imageC1C2C3C3C6C6(C3×He3)⋊C6S3C3×S3C3×S3C32⋊C6C3×C32⋊C6
kernel(C3×He3)⋊C6C33⋊C32C33⋊C6He3⋊4S3C3≀C3C3×He3C1C3×He3He3C33C32C3
# reps116262116212

Matrix representation of (C3×He3)⋊C6 ►in GL18(ℤ)

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

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,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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,-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,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0],[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,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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,-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],[0,0,0,0,0,0,1,-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,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-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,-1,0,0,1,-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] >;
 

(C3×He3)⋊C6 in GAP, Magma, Sage, TeX 

(C_3\times {\rm He}_3)\rtimes C_6
 
% in TeX
 
G:=Group("(C3xHe3):C6");
 
// GroupNames label
 
G:=SmallGroup(486,127);
 
// by ID
 
G=gap.SmallGroup(486,127);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,873,735,3244,3250,11669]);
 
// 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,e*d*e^-1=a*d=d*a,e*a*e^-1=a^-1*c^-1,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=a^-1*b^-1,c*d=d*c,e*c*e^-1=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁