Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #2183 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D7 |
| commutator, socle, prime radical | C22×C7 | C22×C8 |
| Fitting | C7×C4⋊C16 | C2 |
| cosocle | C2×C7×C8 | C23 |
| hypercenter | C4⋊C16 | D7 |
| hypocenter | C7 | #2183 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C8 ◁ C4⋊C16 | |
126 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 896 |
Polycyclic presentation:
2^7,7|a^2=e,b^2=d,c^2=de,e^2=f,f^2=g,b^a=bd,c^a=cg,h^b=h^6,h^c=h^6176 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#2183, C2×#11727036, #13507285, … |
| maximal normal subgroup of | D7×#2183, C2×#11727036, #13507285, … |
| maximal quotient of | #13269142, #13269149, #13269153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 1802);