Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C7 ⋊ #1985 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D7 |
| commutator, socle, prime radical | C23×C7 | C22×C4 |
| Fitting | C7×#336 | C2 |
| cosocle | C22×C4×C7 | C23 |
| hypercenter | #336 | D7 |
| hypocenter | C7 | #1985 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C22×C4 ◁ #336 | |
280 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 896 |
Polycyclic presentation:
2^7,7|a^2=f,b^2=deg,c^2=ef,e^2=g,f^2=g,b^a=bd,c^a=cef,c^b=cg,h^c=h^6164 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#1985, C2×#11726374, #13358249, … |
| maximal normal subgroup of | D7×#1985, C2×#11726374, #13358249, … |
| maximal quotient of | #13253788, #13253790, #13253793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 1140);