Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#3213 ⋊ C2 |
| Sylow | C7 ⋊ #18993 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#6500 |
| commutator, cosocle | C42×C7 | C24 |
| Frattini | C42 | C23×D7 |
| Fitting | C7×#3213 | C2 |
| socle | C22×C7 | C2×#120 |
| hypercenter | #3213 | D7 |
| hypocenter | C7 | #18993 |
| LCS | G ▷ C42×C7 ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #3213 | |
1012 conjugacy classes of subgroups.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^8,7|b^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,i^b=i^6,e^c=eh,f^c=fgh112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 150790);