Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | #41×GL(2,𝔽3) |
| Fitting | Q8×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C24 |
| Frattini, socle | C22 | C23×S4 |
| Fitting | Q8×#41 | D3 |
| hypercenter | C2×#41 | S4 |
| hypocenter | SL(2,𝔽3) | C2×#41 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
1441 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 1441 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | #41×AGL(2,𝔽3), AGL(1,ℤ/8)×GL(2,𝔽3), H(5,𝔽2)×GL(2,𝔽3), … |
| maximal normal subgroup of | AGL(1,ℤ/8)×GL(2,𝔽3), H(5,𝔽2)×GL(2,𝔽3), GL(2,𝔽3)×#41⋊C3 |
| maximal quotient of | #41×AGL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089533);