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) = C24 |
| direct product | C7×#1955 |
| 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 | C22×C7×D4 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
384 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 384 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)(9,11)(13,15)(16,17,21,22,20,18,19,23);(8,9,10,11)(16,18,21,23,20,17,19,22);(12,13,14,15)(16,19,20,21)(17,22,18,23)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5190);