Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C4×A42 |
| unambiguous semidirect product | (C23×C4×A4)⋊C3 |
| unambiguous semidirect product | (C2×C4×C22⋊A4)⋊C3 |
| Fitting, Sylow | C25×C4 ⋊ C32 |
| derived | C24 ⋊ C2×C32×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | A42 |
| commutator, hypocenter | C24 | C2×C32×C4 |
| Frattini | C2 | C22×A42 |
| Fitting | C25×C4 | C32 |
| socle | C26 | C2×C32 |
| prime radical | C22×A42 | C2 |
| cosocle | C25 | C22×C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C4 | |
1028 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 1028 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) = 72 |
128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×A4⋊S4, C2×D4×A42, C2×C4⋊(A4⋊S4), … |
| maximal normal subgroup of | C2×C4×A4⋊S4, C2×D4×A42, C2×C4⋊(A4⋊S4), … |
| maximal quotient of | #72226033, A42×M16, #72228086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157674);