Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D9⋊A4 |
| commutator, hypocenter | Q8×C9 | C3×C4 |
| Frattini | C3×C4 | D3×A4 |
| Fitting, cosocle | C9×#41 | C2×C3 |
| socle | C2×C3 | Dic3×A4 |
| prime radical | C3×#41⋊C3 | D3 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C4 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3,2^3,3,2,3|a^2=c,c^2=g,d^2=g,e^2=g,f^3=h,f^a=f^2h^2,h^a=h^2,d^b=de,e^b=d,f^b=fh,e^d=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | D9⋊(#131⋊C3), #13147141, C2×#11720752, … |
| maximal normal subgroup of | D9⋊(#131⋊C3), #13147141, C2×#11720752, … |
| maximal quotient of | (C9⋊C8)⋊SL(2,𝔽3), C2×#11720752 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 719);