Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C54×C7 · Schur multiplier (H2) = C55 |
| direct product | C52×C7×5−1+2 |
| unambiguous semidirect product | C25⋊(C53×C7) |
| unambiguous semidirect product | (C7×C25)⋊C53 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C53×C7 | C52 |
| commutator, Frattini, cosocle | C5 | C54×C7 |
| prime radical | C54×C7 | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C53×C7 ◁ G | |
2752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2,3,4,5)(36,37,38,39,40,41,42);(6,7,8,9,10);(11,12,14,18,25,13,16,20,27,32,17,23,29,26,33,24,22,19,28,34,31,15,21,30,35);(12,15,22,23,16)(14,19,20,21,29)(18,26,30,27,28)(25,32,33,34,35)5075 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 73);