Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C7×#391 ⋊ C2 |
| Sylow | C7 ⋊ #2808 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C7×#391 | C2 |
| prime radical | C22×D28 | C22 |
| hypercenter | #391 | D7 |
| hypocenter | C7 | #2808 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #391 | |
428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 428 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) = 448 |
Polycyclic presentation:
2^7,7|a^2=e,b^2=f,c^2=fg,d^2=efg,b^a=beg,c^a=cf,d^a=defg,h^a=h^6,c^b=cg,d^b=deg,h^b=h^6116 conjugacy classes — view the full character table.
| maximal subgroup of | #13242559, #13242563, #13242728, … |
| maximal normal subgroup of | #13242559, #13242563, #13242728, … |
| maximal quotient of | #13282259, #13282270, #13282561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 8812);