Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×GO+(4,𝔽2) |
| commutator, cosocle | C3×C3⋊Dic3 | C23 |
| Fitting, socle | C2×C33 | C2×D4 |
| hypocenter | C33 | #135 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 | |
236 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 236 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) = 48 |
Polycyclic presentation:
2^5,3^3|c^2=e,d^2=e,b^a=be,c^a=ce,h^a=h^2,c^b=cd,d^b=de,f^b=f^2,g^b=f^2g,d^c=de,f^c=f^2g^2,f^d=f^2,g^d=g^239 conjugacy classes — view the full character table.
| maximal subgroup of | #13180485, #13180492, #13180500, … |
| maximal normal subgroup of | #13180485, #13180492, #13180500, … |
| maximal quotient of | #13165464, #13165466, #13165486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4100);