Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #10587739⋊C16 |
| derived, Sylow | C137 ⋊ C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #11764049 |
| commutator, hypocenter | C137 | C4×C16 |
| Frattini | C4 | C2×#11764049 |
| Fitting | C8×C137 | C8 |
| socle | C2×C137 | C4×C8 |
| LCS | G ▷ C137 | |
| UCS | C1 ◁ C8 | |
58 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,137|a^2=c,b^2=d,c^2=e,e^2=f,g^a=g^127,g^b=g^37,g^c=g^100,g^d=g^136,g^e=g^136200 conjugacy classes — view the full character table.
| maximal subgroup of | C16×#11764049, #31797804, #31797835, … |
| maximal normal subgroup of | C16×#11764049, #31797804, #31797835, … |
| maximal quotient of | C8×#14555522, #31797807, #31797860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 1642);