Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of (C22⋊C27)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊C27)⋊C8 |
| derived | C22⋊C27 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | (C22⋊C27)⋊C2 |
| commutator, hypocenter | C22⋊C27 | C8 |
| Frattini | C4×C9 | S4 |
| Fitting | C22×C4×C9 | D3 |
| socle, prime radical | C23×C3 | Dic9 |
| cosocle | C4×C22⋊C27 | C2 |
| LCS | G ▷ C22⋊C27 | |
| UCS | C1 ◁ C4 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3^3,2^2|a^2=b,b^2=c,d^3=e,e^3=f,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2,g^a=h,h^a=g,g^d=h,h^d=gh84 conjugacy classes — view the full character table.
| maximal subgroup of | C8×(C22⋊C27)⋊C2, #13136138, C2×(C22⋊C27)⋊C8, … |
| maximal normal subgroup of | C8×(C22⋊C27)⋊C2, #13136138, C2×(C22⋊C27)⋊C8, … |
| maximal quotient of | (Q8⋊C27)⋊C8, (C22⋊C27)⋊C16, #13134133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 74);