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, Frattini, hypercenter | C2 | #89101 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C22 |
| Fitting | C32×Q8 | C2×D3 |
| socle | C2×C32 | C2×S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
204 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 24 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:
2^2,3^3,2^3|f^2=h,g^2=h,b^a=bh,c^a=c^2,d^a=d^2,f^a=gh,g^a=fh,d^b=d^2,e^b=e^2,f^c=g,g^c=fgh,g^f=gh36 conjugacy classes — view the full character table.
| maximal subgroup of | #13180246, #13180247, #13180251, … |
| maximal normal subgroup of | #13180246, #13180247, #13180251, … |
| maximal quotient of | #13166084, #13166086, #13166090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4177);