Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2081 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C22×C7×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
204 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 72 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)(8,9,12,14,13,17,11,16)(10,18,20,23,19,22,21,15);(9,14)(10,19)(11,12)(15,22)(16,17)(18,23);(8,10,11,20,13,19,12,21)(9,15,16,18,17,23,14,22)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5316);