Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×#41⋊SL(2,𝔽3) |
| unambiguous semidirect product | (C3×#41)⋊SL(2,𝔽3) |
| Fitting | C3×Q8×#41 ⋊ C3 |
| Sylow | Q8×#41 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | Q82 | C2×C32 |
| Frattini | C22 | C2×C3×C22⋊A4 |
| Fitting | C3×Q8×#41 | C3 |
| socle | C22×C3 | C2×C22⋊A4 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C2×C3×C4 | |
500 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153843);