Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | D32×C131 ⋊ C2 |
| Sylow | C131 ⋊ D64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #22553147 |
| commutator, cosocle | C32×C131 | C22 |
| Frattini | C32 | C2×D131 |
| Fitting | D32×C131 | C2 |
| socle | C2×C131 | D32 |
| hypercenter | D32 | D131 |
| hypocenter | C131 | D64 |
| LCS | G ▷ C32×C131 ▷ C16×C131 ▷ C8×C131 ▷ C4×C131 ▷ C2×C131 ▷ C131 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ D32 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 8384 |
Polycyclic presentation:
2^7,131|c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^1301270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 2722);