Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C64 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C167⋊C64 |
| derived, Sylow | C167 ⋊ C64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | D167 |
| commutator, hypocenter | C167 | C64 |
| Fitting, cosocle | C32×C167 | C2 |
| socle, prime radical | C2×C167 | C32 |
| LCS | G ▷ C167 | |
| UCS | C1 ◁ C32 | |
14 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,167|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^a=g^1662720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C64×D167, C2×C167⋊C64, #36322348, … |
| maximal normal subgroup of | C64×D167, C2×C167⋊C64, #36322348, … |
| maximal quotient of | C167⋊C128, C2×C167⋊C64 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 268);