direct product, cyclic, abelian, monomial
Aliases: C54, also denoted Z54, SmallGroup(54,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C54 |
| C1 — C54 |
| C1 — C54 |
Generators and relations for C54
G = < a | a54=1 >
Smallest permutation representation of C54
►Regular action on 54 points
Generators in S54
(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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
G:=sub<Sym(54)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)]])
C54 is a maximal subgroup of
Dic27 Q8⋊C27 C7⋊C54
C54 is a maximal quotient of C7⋊C54
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 18A | ··· | 18F | 27A | ··· | 27R | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 |
| kernel | C54 | C27 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 |
Matrix representation of C54 ►in GL1(𝔽109) generated by
| 36 |
G:=sub<GL(1,GF(109))| [36] >;
C54 in GAP, Magma, Sage, TeX
C_{54} % in TeX
G:=Group("C54"); // GroupNames label
G:=SmallGroup(54,2);
// by ID
G=gap.SmallGroup(54,2);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,29,46]);
// Polycyclic
G:=Group<a|a^54=1>;
// generators/relations
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