Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×#226 |
| unambiguous semidirect product | (Q8×#226)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | A4×S4 |
| commutator, cosocle, hypocenter | Q8×SL(2,𝔽3) | C2×C3 |
| Fitting | Q82 | C3×D3 |
| prime radical | SL(2,𝔽3)2 | C2 |
| LCS | G ▷ Q8×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
165 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽3)×SL(2,𝔽7), #226×SL(2,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155456);