Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #6196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×#6200 |
| Sylow | H(3,𝔽3) ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle | C2×H(3,𝔽3) | C24 |
| Frattini | C2×C3 | C22×D32 |
| Fitting | C23×H(3,𝔽3) | C22 |
| socle | C23×C3 | D32 |
| hypocenter | H(3,𝔽3) | C22×D4 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23 | |
960 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | #13140222, #13140256, #13140257, … |
| maximal normal subgroup of | #13140222, #13140256, #13140257, … |
| maximal quotient of | C2×#11721090, C2×#11721094, #13150623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2509);