Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C2×C16 |
| Frattini | C3×C4 | C2×#157 |
| Fitting | C8×H(3,𝔽3) | C4 |
| socle | C2×C3 | #3787 |
| prime radical | C2×#272 | C8 |
| cosocle | #6180 | C22 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C8 | |
86 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3^3|a^2=c,b^2=d,c^2=d,d^2=e,f^a=fg,g^a=fg^2h^2,f^c=f^2,g^c=g^2h,g^f=gh112 conjugacy classes — view the full character table.
| maximal subgroup of | (((C3⋊D3)⋊C3)⋊C2)⋊C16, #13136664, #13136722, … |
| maximal normal subgroup of | (((C3⋊D3)⋊C3)⋊C2)⋊C16, #13136664, #13136722, … |
| maximal quotient of | #13136866, #13136867, #13136908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 737);