Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×#341 ⋊ C2 |
| Sylow | C7 ⋊ #2018 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #6458 |
| commutator, cosocle | C22×C4×C7 | C23 |
| Frattini | C22×C4 | C22×D7 |
| Fitting | C7×#341 | C2 |
| socle | C22×C7 | #121 |
| hypercenter | #341 | D7 |
| hypocenter | C7 | #2018 |
| LCS | G ▷ C22×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #341 | |
382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 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=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^6,c^b=cf,d^b=dg,f^b=fg,h^b=h^6,d^c=dg,e^c=eg,f^c=fg,h^c=h^6110 conjugacy classes — view the full character table.
| maximal subgroup of | #13265928, #13265933, #13265939, … |
| maximal normal subgroup of | #13265928, #13265933, #13265939, … |
| maximal quotient of | #13254397, #13254414, #13254465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 1462);