metacyclic, supersoluble, monomial
Aliases: C9⋊C36, C18.C18, Dic9⋊C9, C9⋊C9⋊C4, (C3×C9).C12, C2.(C9⋊C18), C6.3(S3×C9), C6.4(C9⋊C6), (C3×C18).4C6, (C3×C18).2S3, (C3×Dic9).C3, C3.2(C9⋊C12), C3.3(C9×Dic3), (C3×C9).1Dic3, C32.14(C3×Dic3), (C2×C9⋊C9).C2, (C3×C6).28(C3×S3), SmallGroup(324,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9⋊C36 |
Generators and relations for C9⋊C36
G = < a,b | a9=b36=1, bab-1=a5 >
Smallest permutation representation of C9⋊C36
►On 36 points
Generators in S36
(1 9 29 25 33 17 13 21 5)(2 18 10 14 30 22 26 6 34)(3 23 19 27 11 7 15 35 31)(4 8 24 16 20 36 28 32 12)
(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)
G:=sub<Sym(36)| (1,9,29,25,33,17,13,21,5)(2,18,10,14,30,22,26,6,34)(3,23,19,27,11,7,15,35,31)(4,8,24,16,20,36,28,32,12), (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)>;
G:=Group( (1,9,29,25,33,17,13,21,5)(2,18,10,14,30,22,26,6,34)(3,23,19,27,11,7,15,35,31)(4,8,24,16,20,36,28,32,12), (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) );
G=PermutationGroup([[(1,9,29,25,33,17,13,21,5),(2,18,10,14,30,22,26,6,34),(3,23,19,27,11,7,15,35,31),(4,8,24,16,20,36,28,32,12)], [(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)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 9A | ··· | 9F | 9G | ··· | 9O | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 18G | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | 9 | 9 | 9 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | + | - | |||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | S3 | Dic3 | C3×S3 | C3×Dic3 | S3×C9 | C9×Dic3 | C9⋊C6 | C9⋊C12 | C9⋊C18 | C9⋊C36 |
| kernel | C9⋊C36 | C2×C9⋊C9 | C3×Dic9 | C9⋊C9 | C3×C18 | Dic9 | C3×C9 | C18 | C9 | C3×C18 | C3×C9 | C3×C6 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 1 | 2 | 2 |
Matrix representation of C9⋊C36 ►in GL6(𝔽37)
| 10 | 11 | 27 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 | 0 |
| 9 | 11 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 36 |
| 0 | 0 | 0 | 16 | 36 | 11 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 | 1 | 26 |
| 0 | 0 | 0 | 12 | 27 | 36 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 27 | 36 | 11 | 0 | 0 | 0 |
| 25 | 10 | 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(37))| [10,0,9,0,0,0,11,0,11,0,0,0,27,26,27,0,0,0,0,0,0,1,16,0,0,0,0,0,36,10,0,0,0,36,11,0],[0,0,0,27,25,0,0,0,0,36,10,36,0,0,0,11,1,0,10,12,0,0,0,0,1,27,1,0,0,0,26,36,0,0,0,0] >;
C9⋊C36 in GAP, Magma, Sage, TeX
C_9\rtimes C_{36} % in TeX
G:=Group("C9:C36"); // GroupNames label
G:=SmallGroup(324,9);
// by ID
G=gap.SmallGroup(324,9);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,5404,2170,208,7781]);
// Polycyclic
G:=Group<a,b|a^9=b^36=1,b*a*b^-1=a^5>;
// generators/relations
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