Finite Group DatabasebetaFinite Group Databasebeta| order | 18125 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C29 · Schur multiplier (H2) = C52 |
| direct product | C29×H(3,𝔽5)⋊C5 |
| unambiguous semidirect product | H(3,𝔽5)⋊(C5×C29) |
| unambiguous semidirect product | (C29×H(3,𝔽5))⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C29 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C52×C29 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C29 ◁ C52×C29 ◁ G | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 54 |
| transitive degree | μt(G) = 725 |
A minimal faithful permutation representation — 2 generators on 54 points:
(1,2,4,8,15)(3,5,10,16,22)(6,12,20,25,21)(7,11,19,17,23)(9,18,24,14,13)(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);(2,5,11,13,6)(4,9,10,12,19)(8,16,17,18,20)1421 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18125, 7);