Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#2112 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C23 |
| commutator | C22 | C22×C7×C8 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C22×C7 | C22×C8 |
| prime radical | C24×C7 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
260 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,14,18,15,19,20,25,16,17,21,23,22,24,26,27);(9,11);(13,17)(18,23)(19,24)(25,27)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5347);