Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#456 ⋊ C2 |
| Sylow | C7 ⋊ #3293 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #90281 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#456 | C2 |
| socle | C2×C7 | D42 |
| hypercenter | #456 | D7 |
| hypocenter | C7 | #3293 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #456 | |
444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^7,7|a^2=g,b^2=g,c^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^6,c^b=cg,d^b=de,e^b=eg,h^b=h^6,f^c=fg,e^d=eg77 conjugacy classes — view the full character table.
| maximal subgroup of | #13274558, #13274561, #13274568, … |
| maximal normal subgroup of | #13274558, #13274561, #13274568, … |
| maximal quotient of | #13345102, #13345107, #13345193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 12514);