Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | S4×#226 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S42 |
| commutator, cosocle, hypocenter | A4×SL(2,𝔽3) | C22 |
| Fitting | C22×Q8 | D32 |
| socle | C23 | D3×S4 |
| prime radical | SL(2,𝔽3)×S4 | C2 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
323 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | S4×SL(2,𝔽7), #226×GL(3,𝔽2) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157455);