Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C33 |
No aliases for this group.
| Fitting | C2×C32×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊S4) |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C22 |
| Fitting | C2×C32×Q8 | D3 |
| socle | C2×C32 | C2×S4 |
| hypercenter | C22 | C3⋊(C3⋊S4) |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
442 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 77 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:
2^2,3^3,2^3|f^2=h,g^2=h,b^a=bh,c^a=c^2,d^a=d^2,e^a=e^2,f^a=gh,g^a=fh,f^c=g,g^c=fgh,g^f=gh69 conjugacy classes — view the full character table.
| maximal subgroup of | #13180749, #13180753, C3⋊D3×#983, … |
| maximal normal subgroup of | #13180749, #13180753, C3⋊D3×#983, … |
| maximal quotient of | #13173038, #13173039, #13173044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4435);