Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×H(3,𝔽3) ⋊ D4 |
| Sylow | H(3,𝔽3) ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#1088⋊C2 |
| commutator, cosocle | C2×#272 | C23 |
| Frattini, socle | C2×C3 | C2×GO+(4,𝔽2) |
| Fitting | C22×H(3,𝔽3) | D4 |
| hypercenter | C22 | #1088⋊C2 |
| hypocenter | H(3,𝔽3) | #120 |
| LCS | G ▷ C2×#272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
383 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^5,3^3|b^a=bd,f^a=f^2,g^a=gh,h^a=h^2,c^b=ce,f^b=gh^2,g^b=fh^2,h^b=h^2,f^d=f^2,g^d=g^2h^2,g^f=gh37 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2282);