Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C33 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #89039 |
| commutator, cosocle | C33×C4 | C23 |
| Frattini | C4 | C2×D3×C3⋊D3 |
| Fitting | C33×D4 | C22 |
| socle | C2×C33 | C2×D4 |
| hypercenter | D4 | D3×C3⋊D3 |
| hypocenter | C33 | AGL(1,ℤ/8) |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
624 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 82 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|b^2=e,d^2=e,c^a=cd,d^a=de,h^a=h^2,c^b=cde,d^b=de,f^b=f^2,g^b=g^2,d^c=de66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63937, #13171561, #13171563, … |
| maximal normal subgroup of | D3×#63937, #13171561, #13171563, … |
| maximal quotient of | #13159096, #13159118, #13159143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3531);