Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C3 |
| direct product | #41×C3⋊S4 |
| Fitting | C22×C3×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C3⋊S4 |
| commutator, cosocle | C2×C3×A4 | C24 |
| Frattini | C2 | C23×C3⋊S4 |
| Fitting | C22×C3×#41 | D3 |
| socle | C23×C3 | C23×D3 |
| hypercenter | #41 | C3⋊S4 |
| hypocenter | C3×A4 | C2×#41 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C4 ◁ #41 | |
2034 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 2034 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×C3⋊S4, #41×A4⋊S4, #72274498, … |
| maximal normal subgroup of | AGL(1,ℤ/8)×C3⋊S4, #72274498, D3×#41×S4 |
| maximal quotient of | #41×A4⋊S4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157740);