Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10593100 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C24 |
| Frattini | C2×C3 | C23×S4 |
| Fitting | C2×C3×H(5,𝔽2) | D3 |
| socle | C22×C3 | C22×S4 |
| hypercenter | C2×Q8 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
1385 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 1385 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155419);