Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C7×D4⋊C8 ⋊ C2 |
| Sylow | C7 ⋊ #1621 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #6445 |
| commutator, cosocle | C42×C7 | C23 |
| Frattini | C42 | C22×D7 |
| Fitting | C7×D4⋊C8 | C2 |
| socle | C22×C7 | #120 |
| hypercenter | D4⋊C8 | D7 |
| hypocenter | C7 | #1621 |
| LCS | G ▷ C42×C7 ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ D4⋊C8 | |
278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 55 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|c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^6,d^b=df,e^b=eg,h^b=h^6,d^c=dg,e^c=eg,h^c=h^6110 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#1621, C2×#11725940, #13329695, … |
| maximal normal subgroup of | D7×#1621, C2×#11725940, #13329695, … |
| maximal quotient of | #13242502, #13242514, #13242532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 706);