Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C7×#405 ⋊ C2 |
| Sylow | C7 ⋊ #3155 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D28 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#405 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #405 | D7 |
| hypocenter | C7 | #3155 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 | |
596 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 165 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|b^2=eg,c^2=f,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^6,d^c=dg152 conjugacy classes — view the full character table.
| maximal subgroup of | #13272064, #13272066, #13272678, … |
| maximal normal subgroup of | #13272064, #13272066, #13272678, … |
| maximal quotient of | #13341661, #13341705, #13341726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 10776);