Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C7×#376 ⋊ C2 |
| Sylow | C7 ⋊ #2401 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Frattini | C23 | C23×D7 |
| Fitting | C7×#376 | C2 |
| prime radical | C23×D7 | C23 |
| hypercenter | #376 | D7 |
| hypocenter | C7 | #2401 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
432 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 141 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:
2^7,7|a^2=f,b^2=fg,d^2=e,b^a=be,c^a=cfg,c^b=cg,h^c=h^6164 conjugacy classes — view the full character table.
| maximal subgroup of | #13362050, #13362053, #13362057, … |
| maximal normal subgroup of | #13362050, #13362053, #13362057, … |
| maximal quotient of | #13275491, #13275519, #13275575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5631);