Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C24×GL(2,𝔽3) |
| Fitting | C24×Q8 ⋊ D3 |
| derived | SL(2,𝔽3) ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C25 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C24×Q8 | D3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C25 | |
8957 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
128 conjugacy classes — view the full character table.
| maximal subgroup of | C24×AGL(2,𝔽3), #120×GL(2,𝔽3), GL(2,𝔽3)×C22⋊A4, … |
| maximal normal subgroup of | #120×GL(2,𝔽3), GL(2,𝔽3)×C22⋊A4, C22×A4×GL(2,𝔽3) |
| maximal quotient of | C24×AGL(2,𝔽3), H(5,𝔽2)×GL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090160);