Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #11724023 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C22⋊(C22⋊C9) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C9) | C2×C3 |
| Frattini | C3 | #1020⋊C3 |
| Fitting, socle | C24×C3 | C3×D3 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 | |
79 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3^3,2^4|c^3=d^2,c^a=c^2d,d^a=d^2,e^a=g,f^a=h,g^a=e,h^a=f,e^b=ef,f^b=e,g^b=gh,h^b=g,e^c=f,f^c=ef,g^c=gh,h^c=g30 conjugacy classes — view the full character table.
| maximal subgroup of | #11724023⋊C2, C2×#11724023 |
| maximal normal subgroup of | #11724023⋊C2, C2×#11724023 |
| maximal quotient of | #13164053, #13164057, #13164061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3990);