Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C3×#87547 |
| Fitting | C3×H(7,𝔽2) ⋊ C3 |
| Sylow | H(7,𝔽2) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C32 |
| Frattini | C2 | C22×C3×C22⋊A4 |
| Fitting | C3×H(7,𝔽2) | C3 |
| hypercenter | C3×D4 | C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
2248 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 2248 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#87547, C3×#11715828, C3×#72209831, … |
| maximal normal subgroup of | D3×#87547, C3×#11715828, #72215523, … |
| maximal quotient of | C3×#11923832, ΓL(1,𝔽8)×#87547, #72230283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153915);