Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#447 ⋊ C2 |
| Sylow | C7 ⋊ #3294 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#447 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #447 | D7 |
| hypocenter | C7 | #3294 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #447 | |
494 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^7,7|a^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^6,d^b=de,h^b=h^6,f^c=fg92 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#3294, C2×#11737459, #14230276, … |
| maximal normal subgroup of | D7×#3294, C2×#11737459, #14230276, … |
| maximal quotient of | #13369195, #13370008, #13370935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 12225);