Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #5386 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×C4⋊GL(2,𝔽3) |
| Fitting | C22×C4×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C24 |
| Frattini | C22 | C23×S4 |
| Fitting | C22×C4×Q8 | D3 |
| hypercenter | C23×C4 | S4 |
| hypocenter | SL(2,𝔽3) | C22×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C24 ◁ C23×C4 | |
2309 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 2309 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089109);