direct product, metacyclic, supersoluble, monomial
Aliases: C4xC9:C6, D9:C12, C36:2C6, D18.C6, Dic9:2C6, (C4xD9):C3, C9:C12:2C2, C9:1(C2xC12), C32.(C4xS3), C3.3(S3xC12), C6.13(S3xC6), (C3xC12).8S3, C18.2(C2xC6), (C3xC6).10D6, C12.13(C3xS3), 3- 1+2:1(C2xC4), (C4x3- 1+2):2C2, (C2x3- 1+2).2C22, (C2xC9:C6).C2, C2.1(C2xC9:C6), SmallGroup(216,53)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C4xC9:C6 |
Generators and relations for C4xC9:C6
G = < a,b,c | a4=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C12, D6, C2xC6, C3xS3, C4xS3, C2xC12, S3xC6, C9:C6, S3xC12, C2xC9:C6, C4xC9:C6
Smallest permutation representation of C4xC9:C6
►On 36 points
(1 29 11 20)(2 30 12 21)(3 31 13 22)(4 32 14 23)(5 33 15 24)(6 34 16 25)(7 35 17 26)(8 36 18 27)(9 28 10 19)
(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 11)(2 16 8 10 5 13)(3 12 6 18 9 15)(4 17)(7 14)(19 33 22 30 25 36)(20 29)(21 34 27 28 24 31)(23 35)(26 32)
G:=sub<Sym(36)| (1,29,11,20)(2,30,12,21)(3,31,13,22)(4,32,14,23)(5,33,15,24)(6,34,16,25)(7,35,17,26)(8,36,18,27)(9,28,10,19), (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), (1,11)(2,16,8,10,5,13)(3,12,6,18,9,15)(4,17)(7,14)(19,33,22,30,25,36)(20,29)(21,34,27,28,24,31)(23,35)(26,32)>;
G:=Group( (1,29,11,20)(2,30,12,21)(3,31,13,22)(4,32,14,23)(5,33,15,24)(6,34,16,25)(7,35,17,26)(8,36,18,27)(9,28,10,19), (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), (1,11)(2,16,8,10,5,13)(3,12,6,18,9,15)(4,17)(7,14)(19,33,22,30,25,36)(20,29)(21,34,27,28,24,31)(23,35)(26,32) );
G=PermutationGroup([[(1,29,11,20),(2,30,12,21),(3,31,13,22),(4,32,14,23),(5,33,15,24),(6,34,16,25),(7,35,17,26),(8,36,18,27),(9,28,10,19)], [(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)], [(1,11),(2,16,8,10,5,13),(3,12,6,18,9,15),(4,17),(7,14),(19,33,22,30,25,36),(20,29),(21,34,27,28,24,31),(23,35),(26,32)]])
C4xC9:C6 is a maximal subgroup of
C72:C6 D36:6C6 Dic18:2C6 D36:3C6
C4xC9:C6 is a maximal quotient of C72:C6 Dic9:C12 D18:C12
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 18A | 18B | 18C | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 9 | 9 | 2 | 3 | 3 | 1 | 1 | 9 | 9 | 2 | 3 | 3 | 9 | 9 | 9 | 9 | 6 | 6 | 6 | 2 | 2 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 6 | 6 | 6 | 6 | ··· | 6 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D6 | C3xS3 | C4xS3 | S3xC6 | S3xC12 | C9:C6 | C2xC9:C6 | C4xC9:C6 |
| kernel | C4xC9:C6 | C9:C12 | C4x3- 1+2 | C2xC9:C6 | C4xD9 | C9:C6 | Dic9 | C36 | D18 | D9 | C3xC12 | C3xC6 | C12 | C32 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C4xC9:C6 ►in GL6(F37)
| 31 | 0 | 0 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 31 |
| 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 36 | 36 | 36 | 36 | 36 | 35 |
| 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 0 | 1 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 2 | 1 |
| 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 36 | 36 | 0 |
| 0 | 0 | 36 | 0 | 36 | 0 |
G:=sub<GL(6,GF(37))| [31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[0,0,36,0,0,1,0,0,36,0,0,0,36,36,36,0,1,1,1,0,36,0,0,0,0,0,36,1,0,0,0,0,35,36,1,1],[0,36,1,0,0,0,36,0,1,0,0,0,0,0,1,0,0,36,0,0,1,0,36,0,0,0,2,1,36,36,0,0,1,36,0,0] >;
C4xC9:C6 in GAP, Magma, Sage, TeX
C_4\times C_9\rtimes C_6
% in TeX
G:=Group("C4xC9:C6"); // GroupNames label
G:=SmallGroup(216,53);
// by ID
G=gap.SmallGroup(216,53);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,3604,736,208,5189]);
// Polycyclic
G:=Group<a,b,c|a^4=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations
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