Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of D21 |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = D21 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D21 |
| unambiguous semidirect product | (C2×C3)⋊(C3×D7) |
| Fitting, socle | C2×C32×C7 ⋊ C2 |
| derived | C3×C7 ⋊ C22×C3 |
| Hall | C32×C7 ⋊ C22 |
| Sylow | C32 ⋊ C2×D7 |
| Sylow | C7 ⋊ C2×C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D21 |
| commutator, cosocle, hypocenter | C3×C7 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | C2 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C3 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(4,5)(6,7,8)(10,11)(12,13)(14,15);(1,2)(3,4,5)(9,10,12,14,15,13,11)72 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D21⋊C4, C3×#4246, C3×#4247, … |
| maximal normal subgroup of | C3×D21⋊C4, C3×#4246, C3×#4247, … |
| maximal quotient of | C3×Dic42, C3×C4×D21, C3×D84, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 43);