Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C7×C25 · Schur multiplier (H2) = C52 |
| direct product | C7×#15691075 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | H(3,𝔽5) |
| commutator | C52 | C5×C7×C25 |
| Frattini, cosocle | C53 | C52×C7 |
| prime radical | C54×C7 | C5 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ C53×C7 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 875 |
A minimal faithful permutation representation — 2 generators on 57 points:
(1,2,4,7,11,3,5,8,12,16,6,9,13,17,20,10,14,18,21,23,15,19,22,24,25)(26,27,29,33,40)(28,30,35,41,47)(31,37,45,50,46)(32,36,44,42,48)(34,43,49,39,38)(51,52,53,54,55,56,57);(27,30,36,38,31)(29,34,35,37,44)(33,41,42,43,45)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 18);