Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C53 |
| direct product | C7×C25⋊H(3,𝔽5) |
| unambiguous semidirect product | C25⋊(C7×H(3,𝔽5)) |
| unambiguous semidirect product | (C7×C25)⋊H(3,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | C53 |
| commutator, Frattini, cosocle | C52 | C53×C7 |
| prime radical | C5×C7×H(3,𝔽5) | C5 |
| LCS | G ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ G | |
370 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)(26,27,29,33,40)(28,31,35,42,47)(30,36,45,50,46)(32,38,44,41,48)(34,43,49,39,37)(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);(27,30,37,38,31)(29,34,35,36,44)(33,41,45,42,43)(40,47,48,49,50)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 47);