Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C5 |
| direct product | C7×#15691112 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | C53 |
| commutator, Frattini, cosocle | C52 | C53×C7 |
| prime radical | C53×C7 | C52 |
| LCS | G ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ G | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 3 generators on 57 points:
(2,6,11,9,17)(3,10,12,22,24)(7,18,25,15,19)(8,20,21,23,16)(27,30,28,33,41)(31,32,34,36,37)(38,44,42,40,39)(45,47,49,46,48)(51,52,53,54,55,56,57);(1,2,7,12,16,14,17,25,3,8,5,9,19,22,20,13,11,18,10,21,4,6,15,24,23)(26,27,31,38,45,35,28,32,39,46,50,41,34,40,47,29,30,36,42,48,43,33,37,44,49);(1,3,11,23,25,13,24,17,20,15,14,22,6,16,19,4,12,9,21,7,5,10,2,8,18)(26,28,34,42,49,35,41,36,44,45,50,30,37,38,46,29,33,31,39,47,43,27,32,40,48)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 55);