Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#493 ⋊ C2 |
| Sylow | C7 ⋊ #3274 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#493 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #493 | D7 |
| hypocenter | C7 | #3274 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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|b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,c^b=cg,d^b=de,e^b=eg,h^b=h^6,h^c=h^6,e^d=eg,h^d=h^695 conjugacy classes — view the full character table.
| maximal subgroup of | #13343155, #13343157, #13343200, … |
| maximal normal subgroup of | #13343155, #13343157, #13343200, … |
| maximal quotient of | #13345082, #13345169, #13345285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 13999);