Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×#459 ⋊ C2 |
| Sylow | C7 ⋊ #3122 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#459 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #459 | D7 |
| hypocenter | C7 | #3122 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
418 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^7,7|d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,h^b=h^6,f^c=fg,h^d=h^689 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#3122, C2×#11737762, #14222163, … |
| maximal normal subgroup of | D7×#3122, C2×#11737762, #14222163, … |
| maximal quotient of | #13353889, #13356884, #13356886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 12528);