Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #11720106 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C42⋊C27 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42⋊C27 | C2 |
| Frattini | C22×C9 | S4 |
| Fitting | C42×C9 | D3 |
| socle | C22×C3 | (C22⋊C9)⋊C2 |
| LCS | G ▷ C42⋊C27 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3^3,2^4|b^3=c,c^3=d,e^2=g,f^2=gh,b^a=b^2c^2d^2,c^a=c^2d^2,d^a=d^2,e^a=fh,f^a=eh,g^a=gh,e^b=fg,f^b=ef,g^b=gh,h^b=g42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11720106, #11720106⋊C2 |
| maximal normal subgroup of | C2×#11720106, #11720106⋊C2 |
| maximal quotient of | #13134126, #13134127, #13134128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 73);