Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C7×C25 · Schur multiplier (H2) = C54 |
| direct product | C7×C25×H(3,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center | C5×C7×C25 | C52 |
| commutator | C5 | C52×C7×C25 |
| Frattini, cosocle | C52 | C53×C7 |
| socle | C52×C7 | C53 |
| prime radical | C5×C7×H(3,𝔽5) | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C7×C25 ◁ G | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 3 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)(51,52,53,54,55,56,57);(26,27,29,33,40)(28,31,35,42,47)(30,36,45,50,46)(32,38,44,41,48)(34,43,49,39,37);(27,30,37,38,31)(29,34,35,36,44)(33,41,45,42,43)(40,47,48,49,50)5075 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 43);