Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#1696 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C7 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C7 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ G | |
140 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,15)(10,16,18,23)(11,20,17,14)(13,21,22,19)(24,25,28,31)(26,32,34,39)(27,35,37,30)(29,36,38,33);(8,10,17,21,12,18,11,19)(9,13,20,23,15,22,14,16);(10,18)(13,22)(16,23)(19,21)(24,26,28,34)(25,29,31,38)(27,36,37,33)(30,39,35,32)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4931);