Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C7 ⋊ #1451 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D7 |
| commutator, socle | C22×C7 | C2×C42 |
| Fitting | C4×C7×M16 | C2 |
| prime radical | C23×C7 | C42 |
| cosocle | C42×C7 | C23 |
| hypercenter | C4×M16 | D7 |
| hypocenter | C7 | #1451 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C42 ◁ C4×M16 | |
208 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 121 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=d,b^2=e,d^2=f,e^2=g,b^a=bg,c^a=cf,h^b=h^6176 conjugacy classes — view the full character table.
| maximal subgroup of | #13244363, #13244366, #13244368, … |
| maximal normal subgroup of | #13244363, #13244366, #13244368, … |
| maximal quotient of | #13243156, #13243177, #13244221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 2367);