Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C5 |
| direct product | C7×#15691114 |
| 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 | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 3 generators on 57 points:
(2,4,7,12,20)(5,9,15,8,13)(10,17,14,21,16)(18,25,24,23,22)(27,29,32,37,45)(30,34,40,33,38)(35,42,39,46,41)(43,50,49,48,47)(51,52,53,54,55,56,57);(1,2,5,10,18,3,4,8,14,22,6,7,9,16,23,11,12,13,17,24,19,20,15,21,25);(26,27,30,35,43,28,29,33,39,47,31,32,34,41,48,36,37,38,42,49,44,45,40,46,50)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 57);