Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C7×#120 ⋊ C2 |
| Sylow | C7 ⋊ #2021 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #6500 |
| commutator, cosocle | C24×C7 | C23 |
| Frattini | C24 | C22×D7 |
| Fitting | C2×C7×#120 | C2 |
| socle | C22×C7 | #120 |
| hypercenter | C2×#120 | D7 |
| hypocenter | C7 | #2021 |
| LCS | G ▷ C24×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#120 | |
638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 638 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) = 112 |
Polycyclic presentation:
2^7,7|b^a=bd,c^a=ce,f^a=fg,h^a=h^6,c^b=cf,d^c=dg110 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#2021, C2×#11729425, #13381036, … |
| maximal normal subgroup of | D7×#2021, C2×#11729425, #13381036, … |
| maximal quotient of | #13253989, #13253992, #13253996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4191);