Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C22 |
| Frattini, socle | C2×C3 | D3×S4 |
| Fitting | Q8×C9 | C2×D3 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
131 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 20 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,d^3=e,f^2=h,g^2=h,b^a=bh,d^a=d^2e^2,e^a=e^2,c^b=c^2,f^b=gh,g^b=fh,f^c=g,g^c=fgh,g^f=gh45 conjugacy classes — view the full character table.
| maximal subgroup of | #13164114, C2×#11722234, D9×#983, … |
| maximal normal subgroup of | #13164114, C2×#11722234, D9×#983, … |
| maximal quotient of | #13146208, #13146209, #13146215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2201);