Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C7×#1649 ⋊ C2 |
| Sylow | C7 ⋊ #18516 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D7×#120 |
| commutator, cosocle | C42×C7 | C24 |
| Frattini | C42 | C23×D7 |
| Fitting | C7×#1649 | C2 |
| socle | C22×C7 | C2×#120 |
| hypercenter | #1649 | D7 |
| hypocenter | C7 | #18516 |
| LCS | G ▷ C42×C7 ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1649 | |
1258 conjugacy classes of subgroups.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^8,7|d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,i^a=i^6,e^b=eg,f^b=fh,i^b=i^6,e^c=eh,f^c=fh,i^c=i^6,i^d=i^6130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 145525);