Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C7×#3485 ⋊ C2 |
| Sylow | C7 ⋊ #23902 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#6516 |
| commutator, cosocle | C24×C7 | C24 |
| Frattini | C24 | C23×D7 |
| Fitting | C7×#3485 | C2 |
| socle | C22×C7 | C2×#120 |
| hypercenter | #3485 | D7 |
| hypocenter | C7 | #23902 |
| LCS | G ▷ C24×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #3485 | |
1992 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^8,7|b^2=h,b^a=be,c^a=cf,d^a=de,i^a=i^6,c^b=cg,f^b=fh,e^c=eh,f^d=fh178 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 201165);