Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C7×#123 ⋊ C2 |
| Sylow | C7 ⋊ #2534 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C2×C7×#123 | C2 |
| prime radical | C2×#6500 | C2 |
| hypercenter | C2×#123 | D7 |
| hypocenter | C7 | #2534 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^6,h^c=h^6122 conjugacy classes — view the full character table.
| maximal subgroup of | #13272085, #13272098, #13407185, … |
| maximal normal subgroup of | #13272085, #13272098, #13407185, … |
| maximal quotient of | #13276020, #13276514, #13276571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 15111);