Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#508 ⋊ C2 |
| Sylow | C7 ⋊ #3414 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D7×D8 |
| commutator, cosocle | C7×C8 | C24 |
| Frattini | C8 | C23×D7 |
| Fitting | C7×#508 | C2 |
| socle | C2×C7 | C22×D8 |
| hypercenter | #508 | D7 |
| hypocenter | C7 | #3414 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508 | |
364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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:
2^7,7|d^2=g,e^2=fg,f^2=g,b^a=be,c^a=cg,e^a=ef,f^a=fg,e^b=ef,f^b=fg,h^c=h^698 conjugacy classes — view the full character table.
| maximal subgroup of | #13274467, #13274470, #13274472, … |
| maximal normal subgroup of | #13274467, #13274470, #13274472, … |
| maximal quotient of | #13506697, #13506833, #13507102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 14241);