Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C22×A4×SL(2,𝔽3) |
| Fitting, Sylow | C24×Q8 ⋊ C32 |
| derived | C22×Q8 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | A42 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C32 |
| Frattini | C2 | C22×A42 |
| Fitting | C24×Q8 | C32 |
| socle | C25 | C3×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C23 | |
897 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 897 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | #72228147, C22×A4×GL(2,𝔽3), SL(2,𝔽3)×GL(2,ℤ/4), … |
| maximal normal subgroup of | C22×A4×GL(2,𝔽3), SL(2,𝔽3)×GL(2,ℤ/4), #72244333 |
| maximal quotient of | A4×#2842⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157676);