Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C137⋊#564 |
| Sylow | C137 ⋊ #564 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D137 |
| commutator, socle | C22×C137 | C24 |
| Fitting | #125×C137 | C2 |
| hypercenter | #125 | D137 |
| hypocenter | C137 | #564 |
| LCS | G ▷ C22×C137 ▷ C137 | |
| UCS | C1 ◁ C22 ◁ #125 | |
172 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,137|a^2=e,b^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,g^d=g^136971 conjugacy classes — view the full character table.
| maximal subgroup of | #31780090, #31780092, #31780115, … |
| maximal normal subgroup of | #31780090, #31780092, #31780115, … |
| maximal quotient of | #31783971, #31783972, #31783973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 1304);