Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×GL(2,𝔽3)⋊A4 |
| unambiguous semidirect product | (C2×GL(2,𝔽3))⋊A4 |
| unambiguous semidirect product | (C2×#5924)⋊C3 |
| Fitting | C2×H(5,𝔽2) ⋊ C3×D3 |
| derived | #994 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | A4×S4 |
| commutator, cosocle, hypocenter | #994 | C22×C3 |
| Frattini | C2 | C2×A4×S4 |
| Fitting | C2×H(5,𝔽2) | C3×D3 |
| LCS | G ▷ #994 | |
| UCS | C1 ◁ C22 | |
405 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
58 conjugacy classes — view the full character table.
| maximal subgroup of | #72225978, #72225981, C2×#983⋊A4, … |
| maximal normal subgroup of | #72225978, #72225981, C2×#983⋊A4, … |
| maximal quotient of | #72225978, D3×GL(2,𝔽3)⋊A4, D5×GL(2,𝔽3)⋊A4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157500);