Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#14555522 |
| Sylow | C137 ⋊ C4⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#11764049 |
| commutator | C2×C137 | C2×C16 |
| Fitting | C2×C4×C137 | C8 |
| socle | C22×C137 | C2×C8 |
| hypercenter | C2×C4 | #11764049 |
| hypocenter | C137 | C4⋊C16 |
| LCS | G ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
50 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,137|a^2=d,b^2=c,d^2=e,e^2=f,b^a=bc,g^a=g^127,g^d=g^100,g^e=g^136176 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊#14555522, #31797832, Q8⋊#14555522, … |
| maximal normal subgroup of | D4⋊#14555522, #31797832, Q8⋊#14555522, … |
| maximal quotient of | #31797847, #31797853, #31797859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 1651);