Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2120 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #99 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C2×D4 | C23×C7 |
| socle | C2×C7 | C2×#99 |
| prime radical | C7×#525 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
264 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,14)(11,18,19,13)(12,15)(16,22,23,20)(17,21);(9,13)(14,20)(15,18)(21,22);(8,10,12,17)(9,14,15,21)(11,16,19,23)(13,20,18,22)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5355);