metabelian, supersoluble, monomial, A-group
Aliases: C18.3D6, C6.3D18, Dic3⋊2D9, Dic9⋊2S3, C9⋊S3⋊C4, C6.3S32, C9⋊1(C4×S3), C3⋊1(C4×D9), C2.2(S3×D9), (C3×C6).24D6, (C9×Dic3)⋊3C2, (C3×Dic9)⋊1C2, C32.3(C4×S3), (C3×C18).3C22, (C3×Dic3).5S3, C3.1(C6.D6), (C2×C9⋊S3).C2, (C3×C9)⋊2(C2×C4), SmallGroup(216,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C18.D6 |
Generators and relations for C18.D6
G = < a,b,c | a18=c2=1, b6=a9, bab-1=cac=a-1, cbc=b5 >
Subgroups: 338 in 58 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C9, C9, C32, Dic3, Dic3, C12, D6, D9, C18, C18, C3⋊S3, C3×C6, C4×S3, C3×C9, Dic9, C36, D18, C3×Dic3, C3×Dic3, C2×C3⋊S3, C9⋊S3, C3×C18, C4×D9, C6.D6, C3×Dic9, C9×Dic3, C2×C9⋊S3, C18.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, D9, C4×S3, D18, S32, C4×D9, C6.D6, S3×D9, C18.D6
►On 36 points
(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 29 16 32 13 35 10 20 7 23 4 26)(2 28 17 31 14 34 11 19 8 22 5 25)(3 27 18 30 15 33 12 36 9 21 6 24)
(1 7)(2 6)(3 5)(8 18)(9 17)(10 16)(11 15)(12 14)(19 21)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)
G:=sub<Sym(36)| (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), (1,29,16,32,13,35,10,20,7,23,4,26)(2,28,17,31,14,34,11,19,8,22,5,25)(3,27,18,30,15,33,12,36,9,21,6,24), (1,7)(2,6)(3,5)(8,18)(9,17)(10,16)(11,15)(12,14)(19,21)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36), (1,29,16,32,13,35,10,20,7,23,4,26)(2,28,17,31,14,34,11,19,8,22,5,25)(3,27,18,30,15,33,12,36,9,21,6,24), (1,7)(2,6)(3,5)(8,18)(9,17)(10,16)(11,15)(12,14)(19,21)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)], [(1,29,16,32,13,35,10,20,7,23,4,26),(2,28,17,31,14,34,11,19,8,22,5,25),(3,27,18,30,15,33,12,36,9,21,6,24)], [(1,7),(2,6),(3,5),(8,18),(9,17),(10,16),(11,15),(12,14),(19,21),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30)]])
C18.D6 is a maximal subgroup of
Dic6⋊5D9 Dic18⋊S3 Dic9.D6 C4×S3×D9 D18.3D6 Dic3.D18 D18⋊D6
C18.D6 is a maximal quotient of C36.38D6 C36.40D6 Dic3×Dic9 C18.Dic6 C6.18D36
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 18D | 18E | 18F | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 27 | 27 | 2 | 2 | 4 | 3 | 3 | 9 | 9 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 18 | 18 | 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 | C4 | S3 | S3 | D6 | D6 | D9 | C4×S3 | C4×S3 | D18 | C4×D9 | S32 | C6.D6 | S3×D9 | C18.D6 |
| kernel | C18.D6 | C3×Dic9 | C9×Dic3 | C2×C9⋊S3 | C9⋊S3 | Dic9 | C3×Dic3 | C18 | C3×C6 | Dic3 | C9 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of C18.D6 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 20 | 6 |
| 0 | 0 | 31 | 26 |
| 31 | 6 | 0 | 0 |
| 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 36 | 0 | 0 | 0 |
| 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,20,31,0,0,6,26],[31,31,0,0,6,0,0,0,0,0,0,1,0,0,1,0],[36,36,0,0,0,1,0,0,0,0,0,36,0,0,36,0] >;
C18.D6 in GAP, Magma, Sage, TeX
C_{18}.D_6 % in TeX
G:=Group("C18.D6"); // GroupNames label
G:=SmallGroup(216,28);
// by ID
G=gap.SmallGroup(216,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,31,1065,453,1444,2603]);
// Polycyclic
G:=Group<a,b,c|a^18=c^2=1,b^6=a^9,b*a*b^-1=c*a*c=a^-1,c*b*c=b^5>;
// generators/relations