Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C7 · Schur multiplier (H2) = C5 |
| direct product | C7×#15691061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C52×C7 |
| prime radical | C5×C7×H(3,𝔽5) | C5 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ C53×C7 ◁ G | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 2 generators on 57 points:
(1,2,4,8,15,3,5,10,16,22,7,11,19,17,23,14,13,9,18,24,21,6,12,20,25)(27,29,33,39,31)(30,35,34,38,37)(36,41,45,42,44)(51,52,53,54,55,56,57);(2,5,11,13,6)(4,9,10,12,19)(8,16,17,18,20)(26,27,30,36,43)(28,31,38,44,49)(29,34,41,47,46)(32,39,35,42,48)(33,37,45,50,40)1043 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 4);