Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C7×#392 ⋊ C2 |
| Sylow | C7 ⋊ #1600 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #6500 |
| commutator, cosocle | C22×C4×C7 | C23 |
| Frattini | C22×C4 | C22×D7 |
| Fitting | C7×#392 | C2 |
| socle | C22×C7 | #120 |
| hypercenter | #392 | D7 |
| hypocenter | C7 | #1600 |
| LCS | G ▷ C22×C4×C7 ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #392 | |
320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 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|c^2=fg,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=eg,h^b=h^6,d^c=dg89 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#1600, C2×#11727451, #13326001, … |
| maximal normal subgroup of | D7×#1600, C2×#11727451, #13326001, … |
| maximal quotient of | #13242510, #13242529, #13242540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 2217);