Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C3 |
| direct product | C3×#87536 |
| Fitting | C3×H(7,𝔽2) ⋊ C3 |
| Sylow | H(7,𝔽2) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C24×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C32 |
| Frattini | C2 | C24×C3×A4 |
| Fitting | C3×H(7,𝔽2) | C3 |
| hypercenter | C3×2−1+4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×2−1+4 | |
2744 conjugacy classes of subgroups; only the 538 normal subgroups are shown.
This group has 2744 conjugacy classes of subgroups — too many to draw. Only its 538 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
201 conjugacy classes — view the full character table.
| maximal subgroup of | C3×(2−1+4⋊C3)⋊A4, #72252733, #72252740, … |
| maximal normal subgroup of | C3×(2−1+4⋊C3)⋊A4, #72252733, #72252740, … |
| maximal quotient of | #72245329, #72252733 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153893);