Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C24×C3 |
| direct product | A4×#997 |
| Fitting, derived, Sylow | C22×H(5,𝔽2) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C22×H(5,𝔽2) | C32 |
| socle | C23 | C3×C22⋊A4 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
649 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 649 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
44 conjugacy classes — view the full character table.
| derived subgroup of | #72242980, S4×#5934 |
| maximal subgroup of | A4×B(4,𝔽2)⋊C3, A5×#997, S4×#997, … |
| maximal normal subgroup of | A4×B(4,𝔽2)⋊C3, S4×#997, A4×#5934, … |
| maximal quotient of | SL(2,𝔽3)×#997 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157688);