Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting | C7×#391 ⋊ C2 |
| Sylow | C7 ⋊ #2372 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator | C22×C7 | C23×C4 |
| Fitting | C7×#391 | C2 |
| socle, cosocle | C23×C7 | C24 |
| prime radical | C22×D28 | C22 |
| hypercenter | #391 | D7 |
| hypocenter | C7 | #2372 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #391 | |
560 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 185 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|a^2=g,b^2=eg,c^2=e,d^2=f,b^a=be,c^a=cf,h^a=h^6,d^b=de,h^c=h^6,h^d=h^6134 conjugacy classes — view the full character table.
| maximal subgroup of | #13270628, #13270631, #13271145, … |
| maximal normal subgroup of | #13270628, #13270631, #13271145, … |
| maximal quotient of | #13275247, #13275321, #13275331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 8706);