Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#423 ⋊ C2 |
| Sylow | C7 ⋊ #2931 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D7 |
| commutator, socle | C22×C7 | C23×C4 |
| Fitting | C7×#423 | C2 |
| prime radical | #90168 | C2 |
| cosocle | C2×C4×C7 | C24 |
| hypercenter | #423 | D7 |
| hypocenter | C7 | #2931 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ #423 | |
500 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 189 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,c^2=f,f^2=g,b^a=be,d^a=dg,d^b=dg,h^b=h^6,h^c=h^6140 conjugacy classes — view the full character table.
| maximal subgroup of | #13244798, #13244808, #13244818, … |
| maximal normal subgroup of | #13244798, #13244808, #13244818, … |
| maximal quotient of | #13324311, #13324318, #13324327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 11252);