Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×C8×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7×C8 | C22 |
| commutator | C2 | C2×C4×C7×C8 |
| Frattini, cosocle | C42 | C23×C7 |
| socle | C22×C7 | C2×C42 |
| prime radical | C23×C7 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7×C8 ◁ G | |
224 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 184 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);(16,17,18,19,20,21,22,23);(17,21)(19,23)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4684);