Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C210 |
No aliases for this group.
| Fitting | C22×C7×#120 ⋊ C2 |
| Sylow | C7 ⋊ #15947 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D7 |
| commutator, socle, cosocle | C24×C7 | C24 |
| Fitting | C22×C7×#120 | C2 |
| hypercenter | C22×#120 | D7 |
| hypocenter | C7 | #15947 |
| LCS | G ▷ C24×C7 ▷ C7 | |
| UCS | C1 ◁ C24 ◁ C22×#120 | |
6146 conjugacy classes of subgroups; only the 387 normal subgroups are shown.
This group has 6146 conjugacy classes of subgroups — too many to draw. Only its 387 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|a^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,i^c=i^6226 conjugacy classes — view the full character table.
| maximal subgroup of | #72277148, D7×#15947 |
| maximal normal subgroup of | #72277148, D7×#15947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 117081);