Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C2×C32×Q8 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×(C3⋊D3)⋊A4 |
| commutator, cosocle, hypocenter | C32×Q8 | C22×C3 |
| Frattini, socle | C2×C3 | C2×D3×A4 |
| Fitting | C2×C32×Q8 | C2×C3 |
| hypercenter | C22 | (C3⋊D3)⋊A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 ◁ C22 | |
225 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3,2^2,3^2,2|d^2=h,e^2=h,b^a=bh,f^a=f^2,g^a=g^2,d^c=de,e^c=d,f^c=fg,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #13164197, #13164198, #13164208, … |
| maximal normal subgroup of | #13164197, #13164198, #13164208, … |
| maximal quotient of | #13147212, #13147213, #13147218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2399);