Finite Group DatabasebetaFinite Group Databasebeta| order | 18125 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C29 · Schur multiplier (H2) = C54 |
| direct product | C5×C29×H(3,𝔽5) |
| unambiguous semidirect product | C52⋊(C52×C29) |
| unambiguous semidirect product | (C52×C29)⋊C52 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C29 | C52 |
| commutator, Frattini, cosocle | C5 | C53×C29 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52×C29 ◁ G | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 3625 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59);(6,7,9,13,20)(8,11,15,22,27)(10,16,25,30,26)(12,18,24,21,28)(14,23,29,19,17);(7,10,17,18,11)(9,14,15,16,24)(13,21,25,22,23)(20,27,28,29,30)4205 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18125, 12);