Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of D7×D9 |
| automorphism | Aut(G) = AGL(1,𝔽7)×AGL(1,ℤ/9) (order 2268) · Inn(G) = G · Out(G) = C32 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D7×D9 |
| unambiguous semidirect product | (C7×D9)⋊C2 |
| unambiguous semidirect product | (C9×D7)⋊C2 |
| unambiguous semidirect product | D63⋊C2 |
| Fitting, derived, Hall | C7×C9 ⋊ C22 |
| Sylow | C9 ⋊ C2×D7 |
| Sylow | C7 ⋊ C2×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C9 | C22 |
| Frattini | C3 | D3×D7 |
| socle | C3×C7 | C2×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,4)(3,7)(5,8)(6,9)(10,11,13,15,16,14,12);(1,2,5,3,6,9,7,8,4)(11,12)(13,14)(15,16)30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D7×D9, D7×D27, #67356⋊C2, … |
| maximal normal subgroup of | C2×D7×D9, #67356⋊C2, #67355⋊C2, … |
| maximal quotient of | #92586, D9×Dic7, D63⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 8);