Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | #127 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×#127⋊C3 |
| commutator, cosocle | C3×C42 | C2×C32 |
| Frattini, socle | C22×C3 | C2×C3×A4 |
| Fitting, prime radical | C32×#127 | C3 |
| hypercenter | C32 | #127⋊C3 |
| hypocenter | C42 | C2×3−1+2 |
| LCS | G ▷ C3×C42 ▷ C42 | |
| UCS | C1 ◁ C3 ◁ C32 | |
144 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3^3,2^4|b^3=d,e^2=h,f^2=g,e^a=eh,f^a=fg,c^b=cd,e^b=ef,f^b=eg,g^b=h,h^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #10588306⋊C3, (#63419⋊C2)⋊C3, C2×#11721353, … |
| maximal normal subgroup of | #10588306⋊C3, (#63419⋊C2)⋊C3, C2×#11721353, … |
| maximal quotient of | #13141233, #13141236, #13141248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1320);