Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C7 ⋊ #8136 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C42 | C22×D7 |
| commutator, socle, prime radical | C23×C7 | C2×C42 |
| Fitting | C7×#1273 | C2 |
| cosocle | C2×C42×C7 | C23 |
| hypercenter | #1273 | D7 |
| hypocenter | C7 | #8136 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C42 ◁ #1273 | |
478 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1792 |
Polycyclic presentation:
2^8,7|a^2=e,b^2=f,c^2=d,e^2=g,f^2=h,b^a=bd,c^a=cg,c^b=ch,i^c=i^6328 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 60004);