Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | #64266⋊C3 |
| Fitting | C32×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | (C3⋊S4)⋊C3 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22×C3 |
| Frattini, socle | C2×C3 | C2×C3×S4 |
| Fitting | C32×#41 | D3 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
158 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 25 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|b^2=h,f^2=h,g^2=h,d^a=d^2,e^a=e^2,f^a=gh,g^a=fh,d^c=de,f^d=g,g^d=fgh,g^f=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #13146375, #13146378, #10593638⋊C3, … |
| maximal normal subgroup of | #13146375, #13146378, #10593638⋊C3, … |
| maximal quotient of | C4×(C3⋊#226)⋊C3, #10593623⋊C3, C4×(C3⋊GL(2,𝔽3))⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2325);