Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1891 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #120 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #386 |
| prime radical | C7×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
342 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 108 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)(8,9,12,16,13,10,14,18)(11,17,20,23,19,15,21,22);(8,10,12,18,13,9,14,16)(11,17,21,22,19,15,20,23);(8,11)(9,15)(10,17)(12,21)(13,19)(14,20)(16,23)(18,22)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5126);