Finite Group DatabasebetaFinite Group Databasebeta| order | 225 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C4×C5×GL(2,𝔽3) (order 960) · Inn(G) = C1 · Out(G) = C4×C5×GL(2,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C3 |
| abelian | C32×C25 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C5 | C32×C5 |
| socle, prime radical | C32×C5 | C5 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 225 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31);(4,5,6)225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(225, 2);