Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C32×GL(2,𝔽3) |
| Fitting | C2×C32×Q8 ⋊ D3 |
| derived | SL(2,𝔽3) ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C32 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C22×C32 |
| Frattini | C2 | C2×C32×S4 |
| Fitting | C2×C32×Q8 | D3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C32 | |
362 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 72 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 | C3×#10593203, C3×#10593204, C3×#10593205, … |
| maximal normal subgroup of | C3×#10593203, C3×#10593204, C3×#10593205, … |
| maximal quotient of | C32×#5386, C32×C4×GL(2,𝔽3), C32×C4⋊GL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4412);