Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #41×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C23×C3⋊D3 |
| commutator, cosocle | C2×H(3,𝔽3) | C24 |
| Fitting | #41×H(3,𝔽3) | C2 |
| hypercenter | C3×#41 | C3⋊D3 |
| hypocenter | H(3,𝔽3) | H(5,𝔽2) |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
913 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 913 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^5,3^3|d^2=e,d^a=de,f^a=f^2,g^a=g^2,c^b=ce,g^f=gh91 conjugacy classes — view the full character table.
| maximal subgroup of | #13141078, #13141087, #13144034, … |
| maximal normal subgroup of | #13141078, #13141087, #13144034, … |
| maximal quotient of | #13153888, #13153891, #13153893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2642);