Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C72×C17 · Schur multiplier (H2) = C2×C7 |
| direct product | C72×A4×C17 |
| unambiguous semidirect product | C22⋊(C3×C72×C17) |
| unambiguous semidirect product | (C22×C17)⋊(C3×C72) |
| Fitting, socle, Hall | C22×C72×C17 ⋊ C3 |
| derived, Sylow | C22 ⋊ C3×C72×C17 |
| Hall | C22×C72 ⋊ C3×C17 |
| Hall | C22×C17 ⋊ C3×C72 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C72×C17 | A4 |
| commutator, cosocle, hypocenter | C22 | C3×C72×C17 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72×C17 | C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C72×C17 | |
100 conjugacy classes of subgroups.
3332 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×A4×C17×ΓL(1,𝔽8), A4×C17×#3861, A4×C17×#3862, … |
| maximal normal subgroup of | C7×A4×C17×ΓL(1,𝔽8), A4×C17×#3861, A4×C17×#3862, … |
| maximal quotient of | C72×C17×C22⋊C9, C3×C72×A4×C17 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 171);