Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #63103⋊C7 |
| Fitting, Sylow | #63103 ⋊ C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #90632 |
| commutator, cosocle, hypocenter | H(7,𝔽2) | C2×C7 |
| Frattini, socle | C2 | C2×#90632 |
| Fitting | #63103 | C7 |
| LCS | G ▷ H(7,𝔽2) | |
| UCS | C1 ◁ C4 | |
4267 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 4267 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
7,2^8|h^2=i,b^a=cfgh,c^a=cdeghi,d^a=bi,e^a=cefhi,f^a=e,g^a=bcg,c^b=ci,g^b=gi,f^c=fi,e^d=ei46 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1083546);