Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C22×Ω+(4,𝔽3) |
| Fitting, Sylow | C22×H(5,𝔽2) ⋊ C32 |
| derived | H(5,𝔽2) ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | A42 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C32 |
| Frattini | C2 | C22×A42 |
| Fitting | C22×H(5,𝔽2) | C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23 | |
727 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 727 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | C22×SO+(4,𝔽3), C23×Ω+(4,𝔽3), #72225973, … |
| maximal normal subgroup of | C22×SO+(4,𝔽3), C23×Ω+(4,𝔽3), #72225973, … |
| maximal quotient of | C23×Ω+(4,𝔽3), D4×Ω+(4,𝔽3), #72228147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157679);