Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| Fitting | C7×#1721 ⋊ C2 |
| Sylow | C7 ⋊ #19745 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D7×#127 |
| commutator, cosocle | C42×C7 | C24 |
| Frattini | C42 | C23×D7 |
| Fitting | C7×#1721 | C2 |
| socle | C22×C7 | C2×#127 |
| hypercenter | #1721 | D7 |
| hypocenter | C7 | #19745 |
| LCS | G ▷ C42×C7 ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1721 | |
994 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^8,7|b^2=e,c^2=f,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg,d^c=dh,i^c=i^6,i^d=i^6118 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#19745, #72259644, #72259889 |
| maximal normal subgroup of | D7×#19745, #72259644, #72259889 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 158808);