Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | S4×#41⋊C3 |
| unambiguous semidirect product | (#41×S4)⋊C3 |
| Fitting | C22×#41 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | A4×S4 |
| commutator, cosocle, hypocenter | Q8×A4 | C22×C3 |
| Frattini | C2 | C2×A4×S4 |
| Fitting | C22×#41 | C3×D3 |
| socle | C23 | C2×D3×A4 |
| LCS | G ▷ Q8×A4 | |
| UCS | C1 ◁ C4 | |
446 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#6732, S4×GU(2,𝔽3) |
| maximal normal subgroup of | S4×GU(2,𝔽3) |
| maximal quotient of | GL(2,𝔽3)×#41⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157495);