Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of S4×C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C25 |
| direct product | S4×C22⋊A4 |
| unambiguous semidirect product | A4⋊(C2×C22⋊A4) |
| Fitting, socle | C26 ⋊ C3×D3 |
| derived | C24×A4 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24×A4 | C2×C3 |
| Fitting, socle | C26 | C3×D3 |
| LCS | G ▷ C24×A4 | |
| UCS | C1 | |
1868 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 1868 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | A42×S4, S4×#1020, C22⋊A4×#1020, … |
| maximal normal subgroup of | A42×S4, S4×#1020, S4×#120⋊C3 |
| maximal quotient of | C22⋊A4×#1020, GL(2,𝔽3)×C22⋊A4, S4×#997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157856);