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) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic21 |
| unambiguous semidirect product | C3⋊(C3×Dic7) |
| derived | C3×C7 ⋊ C3×C4 |
| Hall | C32×C7 ⋊ C4 |
| Sylow | C32 ⋊ Dic7 |
| Sylow | C7 ⋊ C3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D21 |
| commutator, hypocenter | C3×C7 | C3×C4 |
| Frattini | C2 | C3×D21 |
| Fitting, socle, prime radical | C2×C32×C7 | C2 |
| cosocle | C2×C3×C7 | C2×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C3 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3)(4,5)(6,7)(8,9,10)(12,13)(14,17,16,15);(1,2,4,6,7,5,3)(11,12,13)72 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic3×D7, C3×D3×Dic7, C3×#4245, … |
| maximal normal subgroup of | C3×Dic3×D7, C3×D3×Dic7, C3×#4245, … |
| maximal quotient of | C3×C3⋊(C7⋊C8), C2×C3×Dic21, C3×Dic63, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 22);