Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C128 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊C128 |
| derived, Sylow | C7 ⋊ C128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C64 | D7 |
| commutator, hypocenter | C7 | C128 |
| Fitting, cosocle | C7×C64 | C2 |
| socle, prime radical | C2×C7 | C64 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C64 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 896 |
Polycyclic presentation:
2^7,7|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^6320 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C128, C2×C7⋊C128, #13241706, … |
| maximal normal subgroup of | D7×C128, C2×C7⋊C128, #13241706, … |
| maximal quotient of | C7⋊C256, C2×C7⋊C128 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 1);