Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×C3×#997 |
| Fitting | C22×C3×H(5,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C22×C32 |
| Sylow | C22×H(5,𝔽2) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C3 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C32 |
| Frattini | C2 | C22×C3×C22⋊A4 |
| Fitting | C22×C3×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23×C3 | |
2608 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 2608 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153909);