Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C23×C3×#226 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24×C3 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24×C3 |
| Frattini | C2 | C23×C3×S4 |
| Fitting | C23×C3×Q8 | D3 |
| prime radical | C23×C3×SL(2,𝔽3) | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C24×C3 | |
1689 conjugacy classes of subgroups; only the 300 normal subgroups are shown.
This group has 1689 conjugacy classes of subgroups — too many to draw. Only its 300 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157700);