Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3×D3×GL(2,𝔽3) (order 1728) · Inn(G) = C1 · Out(G) = C2×C3×D3×GL(2,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C3 |
| abelian | C22×C32×C7 |
| multiplicative | (ℤ/301)ˣ, (ℤ/387)ˣ, (ℤ/441)ˣ, (ℤ/602)ˣ, (ℤ/774)ˣ, (ℤ/882)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2)(5,6,7)(11,12,13,14,15,16,17);(3,4)(8,9,10)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 46);