Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×A4⋊GL(2,𝔽3) |
| unambiguous semidirect product | (C2×A4)⋊GL(2,𝔽3) |
| Fitting | C23×Q8 ⋊ C3⋊D3 |
| derived | A4×SL(2,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4⋊S4 |
| commutator, cosocle, hypocenter | A4×SL(2,𝔽3) | C22 |
| Frattini | C2 | C2×A4⋊S4 |
| Fitting | C23×Q8 | C3⋊D3 |
| socle | C24 | C3⋊S4 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
543 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 543 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×S4×GL(2,𝔽3), #72241503, C2×#72213290, … |
| maximal normal subgroup of | C2×S4×GL(2,𝔽3), #72244327, #72244333 |
| maximal quotient of | #72241503, C2×#72213290, #72244333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157522);