Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | SL(2,𝔽3)×#41⋊C3 |
| unambiguous semidirect product | (#41×SL(2,𝔽3))⋊C3 |
| unambiguous semidirect product | (Q8×#41⋊C3)⋊C3 |
| unambiguous semidirect product | (#41⋊SL(2,𝔽3))⋊C3 |
| Fitting, Sylow | Q8×#41 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | A42 |
| commutator, cosocle, hypocenter | Q82 | C2×C32 |
| Frattini, socle | C22 | C2×A42 |
| Fitting | Q8×#41 | C32 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C2×C4 | |
242 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
98 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽3)×#6732, SL(2,𝔽3)×GU(2,𝔽3), #72276811, … |
| maximal normal subgroup of | SL(2,𝔽3)×GU(2,𝔽3), #72276811, GL(2,𝔽3)×#41⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156571);