Finite Group DatabasebetaFinite Group Databasebeta| order | 18125 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C25×C29 · Schur multiplier (H2) = C5 |
| direct product | C29×C25⋊C25 |
| unambiguous semidirect product | C25⋊(C25×C29) |
| unambiguous semidirect product | (C25×C29)⋊C25 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C52×C29 | C52 |
| commutator | C5 | C5×C25×C29 |
| Frattini, cosocle | C52 | C52×C29 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52×C29 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 18125 |
A minimal faithful permutation representation — 2 generators on 79 points:
(2,4,7,12,20)(5,9,15,8,13)(10,17,14,21,16)(18,25,24,23,22)(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);(1,2,5,10,18,3,4,8,14,22,6,7,9,16,23,11,12,13,17,24,19,20,15,21,25)4205 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18125, 4);