Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Sylow | C7 ⋊ #3502 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D7 |
| commutator, socle, cosocle | C22×C7 | C25 |
| Fitting | C2×C4×C7×Q8 | C2 |
| prime radical | C23×C7 | C24 |
| hypercenter | C2×C4×Q8 | D7 |
| hypocenter | C7 | #3502 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 | |
964 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 553 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=g,b^2=f,c^2=g,d^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^6,d^b=df164 conjugacy classes — view the full character table.
| maximal subgroup of | #13245358, #13245361, #13245795, … |
| maximal normal subgroup of | #13245358, #13245361, #13245795, … |
| maximal quotient of | #13513893, #13514236, #13514734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17637);