Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#399 ⋊ C2 |
| Sylow | C7 ⋊ #2437 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C7×#399 | C2 |
| prime radical | C24×D7 | C22 |
| hypercenter | #399 | D7 |
| hypocenter | C7 | #2437 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #399 | |
780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 780 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|a^2=f,c^2=f,b^a=be,c^a=cf,d^a=de,h^a=h^6,c^b=cg,h^c=h^6,h^d=h^6104 conjugacy classes — view the full character table.
| maximal subgroup of | #13265995, #13265999, #13266581, … |
| maximal normal subgroup of | #13265995, #13265999, #13266581, … |
| maximal quotient of | #13275751, #13276710, #13276996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 10313);