direct product, cyclic, abelian, monomial
Aliases: C36, also denoted Z36, SmallGroup(36,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C36 |
| C1 — C36 |
| C1 — C36 |
Generators and relations for C36
G = < a | a36=1 >
Smallest permutation representation of C36
►Regular action on 36 points
Generators in S36
(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)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)]])
C36 is a maximal subgroup of
C9⋊C8 Dic18 D36 Q8.C18 C7⋊C36 He3.3C12 C13⋊2C36 C13⋊C36
C36 is a maximal quotient of
C7⋊C36 C13⋊2C36 C13⋊C36
| action | f(x) | Disc(f) |
|---|---|---|
| 36T1 | x36+x35+x34+x33+x32+x31+x30+x29+x28+x27+x26+x25+x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1 | 3735 |
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 36A | ··· | 36L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 |
| kernel | C36 | C18 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 |
Matrix representation of C36 ►in GL1(𝔽37) generated by
| 17 |
G:=sub<GL(1,GF(37))| [17] >;
C36 in GAP, Magma, Sage, TeX
C_{36} % in TeX
G:=Group("C36"); // GroupNames label
G:=SmallGroup(36,2);
// by ID
G=gap.SmallGroup(36,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-3,24,53]);
// Polycyclic
G:=Group<a|a^36=1>;
// generators/relations
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