Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#1851 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C23 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C23×C7 | C22×C4 |
| prime radical | C24×C7 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
324 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13)(14,15)(16,17,20,18,19,21,23,22);(9,11)(16,18,20,21,19,22,23,17);(13,15)(17,21)(18,22)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5086);