Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#472 ⋊ C2 |
| Sylow | C7 ⋊ #3067 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7×#121 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#472 | C2 |
| socle | C2×C7 | C2×#121 |
| hypercenter | #472 | D7 |
| hypocenter | C7 | #3067 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #472 | |
452 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 123 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=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^b=h^6,f^c=fg,h^d=h^680 conjugacy classes — view the full character table.
| maximal subgroup of | #13507747, #13507749, #13507758, … |
| maximal normal subgroup of | #13507747, #13507749, #13507758, … |
| maximal quotient of | #13336036, #13336068, #13336101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 13148);