Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C3×#5950 |
| Fitting | C2×C3×#585 ⋊ C3 |
| Sylow | C2×#585 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C32 |
| Frattini | C2 | C22×C3×C22⋊A4 |
| Fitting | C2×C3×#585 | C3 |
| socle | C22×C3 | C2×C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3×C4 | |
2236 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 2236 conjugacy classes of subgroups — too many to draw. Only its 96 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.
| maximal subgroup of | AGL(1,𝔽7)×#5950, C2×D3×#5950, C2×C3×#87519, … |
| maximal normal subgroup of | C2×D3×#5950, C2×C3×#87519, C2×#11923784, … |
| maximal quotient of | AGL(1,𝔽7)×#5950 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153913);