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×#2082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C23×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 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) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,18,14,19,12,17)(10,20,23,15,24,31,22,32)(11,25,27,33,28,34,29,16)(21,36,37,39,26,30,38,35);(8,10,14,24)(9,15,19,32)(11,26,28,21)(12,23,13,22)(16,30,33,36)(17,31,18,20)(25,39,34,35)(27,37,29,38);(8,11,12,27,14,28,13,29)(9,16,17,25,19,33,18,34)(10,21,22,37,24,26,23,38)(15,30,20,35,32,36,31,39)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5317);