Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
No aliases for this group.
| Fitting | C7×2−1+6 ⋊ C2 |
| Sylow | C7 ⋊ #63103 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C26×D7 |
| commutator, socle, cosocle | C2×C7 | C27 |
| Fitting | C7×2−1+6 | C2 |
| hypercenter | 2−1+6 | D7 |
| hypocenter | C7 | #63103 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ 2−1+6 | |
59452 conjugacy classes of subgroups; only the 32039 normal subgroups are shown.
This group has 59452 conjugacy classes of subgroups — too many to draw. Only its 32039 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^8,7|g^2=h,b^a=bh,f^a=fh,e^b=eh,d^c=dh,i^e=i^6,i^f=i^6,i^g=i^6325 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1083471);