Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C7 ⋊ #23359 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C23×D7 |
| commutator, socle | C23×C7 | C23×C4 |
| Fitting, prime radical | C7×#2397 | C2 |
| cosocle | C22×C4×C7 | C24 |
| hypercenter | #2397 | D7 |
| hypocenter | C7 | #23359 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C22×C4 ◁ #2397 | |
882 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 882 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 896 |
Polycyclic presentation:
2^8,7|a^2=g,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=defh,i^a=i^6,c^b=cefh,d^b=de,d^c=dh184 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 196133);