Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C16 | C3⋊D3 |
| commutator, hypocenter | H(3,𝔽3) | C32 |
| Fitting, cosocle | C16×H(3,𝔽3) | C2 |
| socle | C2×C3 | C3⋊(C3⋊C16) |
| prime radical | C2×H(3,𝔽3) | C16 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C16 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^5,3^3|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^2,g^a=g^2,g^f=gh160 conjugacy classes — view the full character table.
| maximal subgroup of | #13134890, ((C3⋊D3)⋊C3)⋊C32, #13134910, … |
| maximal normal subgroup of | #13134890, ((C3⋊D3)⋊C3)⋊C32, #13134910, … |
| maximal quotient of | #13134122, C2×#11720102, C3×#11720102 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 69);