Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C7×2−1+6 ⋊ C2 |
| Sylow | C7 ⋊ #63093 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×C22⋊D7 |
| commutator, cosocle | C4×C7 | C26 |
| Frattini | C4 | C25×D7 |
| Fitting | C7×2−1+6 | C2 |
| socle | C2×C7 | C24×D4 |
| hypercenter | 2−1+6 | D7 |
| hypocenter | C7 | #63093 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ 2−1+6 | |
10698 conjugacy classes of subgroups.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^8,7|a^2=h,b^2=h,g^2=h,b^a=bg,g^a=gh,g^b=gh,i^b=i^6,f^c=fh,e^d=eh277 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1083439);