Finite Group DatabasebetaFinite Group Databasebeta| order | 552 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C23 · Schur multiplier (H2) = C2 |
| direct product | C2×A4×C23 |
| unambiguous semidirect product | C23⋊(C3×C23) |
| unambiguous semidirect product | (C23×C23)⋊C3 |
| Fitting, socle, Hall | C23×C23 ⋊ C3 |
| derived | C22 ⋊ C2×C3×C23 |
| Sylow | C23 ⋊ C3×C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C23 | A4 |
| commutator, cosocle, hypocenter | C22 | C2×C3×C23 |
| Frattini | C1 | G |
| Fitting, socle | C23×C23 | C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C23 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 138 |
184 conjugacy classes — view the full character table.
| maximal subgroup of | C23×SL(2,ℤ/4), A4⋊Dic23, A4×Dic23, … |
| maximal normal subgroup of | C23×SL(2,ℤ/4), A4⋊Dic23, A4×Dic23, … |
| maximal quotient of | C4×A4×C23, C2×C23×SL(2,𝔽3), C23×#41⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(552, 35);