Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#504 ⋊ C2 |
| Sylow | C7 ⋊ #3415 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D56 |
| commutator, cosocle | C7×C8 | C24 |
| Frattini | C8 | C23×D7 |
| Fitting | C7×#504 | C2 |
| socle | C2×C7 | C22×D8 |
| hypercenter | #504 | D7 |
| hypocenter | C7 | #3415 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 | |
370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 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|e^2=fg,f^2=g,b^a=be,e^a=ef,f^a=fg,h^a=h^6,e^b=ef,f^b=fg,h^b=h^6,d^c=dg152 conjugacy classes — view the full character table.
| maximal subgroup of | #13274478, #13274482, #13274492, … |
| maximal normal subgroup of | #13274478, #13274482, #13274492, … |
| maximal quotient of | #13506709, #13506840, #13507140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 14224);