Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C22×C8 |
| direct product | C2×C8×AGL(1,𝔽9) |
| Fitting | C2×C32×C8 ⋊ C8 |
| derived, Sylow | C32 ⋊ C2×C82 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C2×C82 |
| Frattini | C4 | C22×AGL(1,𝔽9) |
| Fitting | C2×C32×C8 | C8 |
| socle | C22×C32 | C4×C8 |
| prime radical | C22×C3⋊D3 | C42 |
| cosocle | C4×#157 | C23 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C8 | |
318 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C8×AΓL(1,𝔽9), C2×#72220215, #72252779, … |
| maximal normal subgroup of | C2×C8×AΓL(1,𝔽9), #72252779, M32×AGL(1,𝔽9), … |
| maximal quotient of | C2×#72220215, M32×AGL(1,𝔽9), C2×AGL(1,𝔽9)2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 7512);