Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×#3517⋊C3 |
| unambiguous semidirect product | #3517⋊C32 |
| unambiguous semidirect product | (C3×#3517)⋊C3 |
| Fitting | C3×#3517 ⋊ C3 |
| Sylow | #3517 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C32 |
| Frattini | C22 | C23×C3×A4 |
| Fitting | C3×#3517 | C3 |
| hypercenter | C3×#125 | A4 |
| hypocenter | Q8 | C32×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#125 | |
630 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153771);