Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C7 ⋊ #2264 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#6400 |
| commutator, cosocle | C7×C16 | C23 |
| Frattini | C16 | C22×D7 |
| Fitting, prime radical | C7×#508 | C2 |
| socle | C2×C7 | C2×D16 |
| hypercenter | #508 | D7 |
| hypocenter | C7 | #2264 |
| LCS | G ▷ C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #508 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^7,7|b^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,d^b=de,e^b=ef,f^b=fg,h^b=h^692 conjugacy classes — view the full character table.
| maximal subgroup of | #13274411, #13274414, #13274474, … |
| maximal normal subgroup of | #13274411, #13274414, #13274474, … |
| maximal quotient of | #13274235, #13274242, #13274304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4030);