C54 - GroupNames

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Gn Go

G = C54  order 54 = 2·33

Cyclic group

  • Order 54 #2
  • ← prev ←
  • → next →
  • Label
  • C54
  • Frattini
  • Φ=C9
  • G/Φ=C6
    • 2-Sylow◁GC2 3-Sylow◁GC27
  • Aut G = Out G
  • C18

direct product, cyclic, abelian, monomial

Aliases: C54, also denoted Z54, SmallGroup(54,2)

Series: Derived ►Chief ►Lower central ►Upper central

Derived seriesC1 — C54
Chief seriesC1 — C3 — C9 — C27 — C54
Lower centralC1 — C54
Upper centralC1 — C54

Generators and relations for C54  G = < a | a54=1 >

C1 C2 C3 C6 C9 C18 C27 C54 Smallest permutation representation of C54 ►Regular action on 54 pointsGenerators 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) Copy 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)>; Copy Magma code 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) ); Copy GAP code 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)]]) Copy Sage code

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 3A3B6A6B9A···9F18A···18F27A···27R54A···54R
order1233669···918···1827···2754···54
size1111111···11···11···11···1

54 irreducible representations

dim11111111
type++
imageC1C2C3C6C9C18C27C54
kernelC54C27C18C9C6C3C2C1
# reps1122661818

Matrix representation of C54 ►in GL1(𝔽109) generated by

36
G:=sub<GL(1,GF(109))| [36] >; Copy Magma code

C54 in GAP, Magma, Sage, TeX C_{54} % in TeX  Copy TeX code G:=Group("C54"); // GroupNames label  To be in Magma G:=SmallGroup(54,2); // by ID  Copy Magma/GAP code G=gap.SmallGroup(54,2); # by ID  Copy Sage code G:=PCGroup([4,-2,-3,-3,-3,29,46]); // Polycyclic  Copy Magma code G:=Group<a|a^54=1>; // generators/relations  Copy Magma code

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Subgroup lattice of C54 in TeX

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