direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×S3×D9, C6⋊1D18, C18⋊1D6, C6.8S32, (C3×C9)⋊C23, C9⋊S3⋊C22, (C3×S3).D6, (C6×D9)⋊5C2, (S3×C9)⋊C22, (C3×C18)⋊C22, (C3×D9)⋊C22, (S3×C18)⋊5C2, (S3×C6).4S3, C9⋊1(C22×S3), (C3×C6).29D6, C3⋊1(C22×D9), C32.2(C22×S3), C3.1(C2×S32), (C2×C9⋊S3)⋊5C2, SmallGroup(216,101)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C2×S3×D9 |
Generators and relations for C2×S3×D9
G = < a,b,c,d,e | a2=b3=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 622 in 106 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, S3, S3, C6, C6, C23, C9, C9, C32, D6, D6, C2×C6, D9, D9, C18, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C22×S3, C3×C9, D18, D18, C2×C18, S32, S3×C6, S3×C6, C2×C3⋊S3, C3×D9, S3×C9, C9⋊S3, C3×C18, C22×D9, C2×S32, S3×D9, C6×D9, S3×C18, C2×C9⋊S3, C2×S3×D9
Quotients: C1, C2, C22, S3, C23, D6, D9, C22×S3, D18, S32, C22×D9, C2×S32, S3×D9, C2×S3×D9
►On 36 points
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 19)(8 20)(9 21)(10 36)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)
(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)(28 29 30 31 32 33 34 35 36)
(1 21)(2 20)(3 19)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)
G:=sub<Sym(36)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,19)(8,20)(9,21)(10,36)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35), (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)(28,29,30,31,32,33,34,35,36), (1,21)(2,20)(3,19)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,19)(8,20)(9,21)(10,36)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35), (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)(28,29,30,31,32,33,34,35,36), (1,21)(2,20)(3,19)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,19),(8,20),(9,21),(10,36),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35)], [(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),(28,29,30,31,32,33,34,35,36)], [(1,21),(2,20),(3,19),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,36),(18,35)]])
C2×S3×D9 is a maximal subgroup of
C36⋊D6 D18⋊D6
C2×S3×D9 is a maximal quotient of
D18.D6 Dic6⋊5D9 Dic18⋊S3 D12⋊5D9 D12⋊D9 D6.D18 D36⋊5S3 Dic9.D6 C36⋊D6 D18.3D6 Dic3.D18 D18.4D6 D18⋊D6
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 3 | 3 | 9 | 9 | 27 | 27 | 2 | 2 | 4 | 2 | 2 | 4 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D9 | D18 | D18 | S32 | C2×S32 | S3×D9 | C2×S3×D9 |
| kernel | C2×S3×D9 | S3×D9 | C6×D9 | S3×C18 | C2×C9⋊S3 | D18 | S3×C6 | D9 | C18 | C3×S3 | C3×C6 | D6 | S3 | C6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 6 | 3 | 1 | 1 | 3 | 3 |
Matrix representation of C2×S3×D9 ►in GL4(𝔽19) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 1 | 0 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 5 |
| 0 | 0 | 14 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 17 | 14 |
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[18,18,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,2,14,0,0,5,7],[1,0,0,0,0,1,0,0,0,0,5,17,0,0,12,14] >;
C2×S3×D9 in GAP, Magma, Sage, TeX
C_2\times S_3\times D_9
% in TeX
G:=Group("C2xS3xD9"); // GroupNames label
G:=SmallGroup(216,101);
// by ID
G=gap.SmallGroup(216,101);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,1065,453,1444,2603]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^9=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations