Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C27×C4 |
| direct product | C7×#3549 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ G | |
1258 conjugacy classes of subgroups; only the 768 normal subgroups are shown.
This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 768 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) = 224 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3,4,5,6,7)(9,11)(13,14)(17,20)(18,19);(8,9,12,11)(10,14,15,13)(16,17,21,20)(18,23,19,22);(16,18,21,19)(17,22,20,23);(8,10,12,15)(9,13,11,14)(16,19,21,18)(17,23,20,22);(8,11,12,9)(10,14,15,13)(16,20,21,17)(18,23,19,22)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 19267);