Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×(C22⋊C27)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C27 | C22 |
| Frattini | C2×C9 | C2×S4 |
| Fitting | C2×Q8×C9 | D3 |
| socle | C2×C3 | C2×(C22⋊C9)⋊C2 |
| hypercenter | C22 | (C22⋊C27)⋊C2 |
| LCS | G ▷ Q8⋊C27 | |
| UCS | C1 ◁ C2 ◁ C22 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^2,3^3,2^3|c^3=d,d^3=e,f^2=h,g^2=h,b^a=bh,c^a=c^2d^2e^2,d^a=d^2e^2,e^a=e^2,f^a=gh,g^a=fh,f^c=g,g^c=fgh,g^f=gh69 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11720649, #13145305, #13145308, … |
| maximal normal subgroup of | C2×#11720649, #13145305, #13145308, … |
| maximal quotient of | #13136126, #13136127, #13136132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 616);