Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#15691108 |
| 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:
(2,6,14,19,8)(3,9,20,23,11)(7,16,18,15,24)(10,21,22,17,25)(27,28,32,39,48)(30,33,40,31,37)(35,41,36,45,38)(43,49,47,46,44)(51,52,53,54,55,56,57);(1,2,7,17,9,5,8,18,21,20,13,19,24,25,23,12,14,16,22,11,4,6,15,10,3)(26,27,30,35,43,29,28,31,36,44,34,32,33,38,46,42,39,37,41,47,50,48,40,45,49);(1,3,10,15,6,4,11,22,16,14,12,23,25,24,19,13,20,21,18,8,5,9,17,7,2)(26,28,33,41,49,29,32,37,45,43,34,39,40,35,44,42,48,30,36,46,50,27,31,38,47)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 51);