Finite Group DatabasebetaFinite Group Databasebeta| order | 18125 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C25×C29 · Schur multiplier (H2) = C52 |
| direct product | C29×C52⋊C25 |
| unambiguous semidirect product | C52⋊(C25×C29) |
| unambiguous semidirect product | (C52×C29)⋊C25 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C29 | C52 |
| commutator | C5 | C5×C25×C29 |
| Frattini, cosocle | C52 | C52×C29 |
| prime radical | C53×C29 | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52×C29 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 3625 |
A minimal faithful permutation representation — 2 generators on 79 points:
(1,2,4,8,15)(3,6,10,17,22)(5,11,20,25,21)(7,13,19,16,23)(9,18,24,14,12)(26,27,29,32,36,28,30,33,37,41,31,34,38,42,45,35,39,43,46,48,40,44,47,49,50)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(2,5,12,13,6)(4,9,10,11,19)(8,16,20,17,18)(15,22,23,24,25)4205 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18125, 3);