Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×#10596399 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | A4⋊S4 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C22 |
| Frattini | C2 | C2×A4⋊S4 |
| Fitting | C2×H(5,𝔽2) | C3⋊D3 |
| prime radical | C2×Ω+(4,𝔽3) | C2 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C22 | |
377 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11923520, #72226061, #72226080, … |
| maximal normal subgroup of | C2×#11923520, #72226061, #72226080, … |
| maximal quotient of | #72226080, C4×#10596399, C2×#11921471 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157527);