Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2050 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #394 |
| prime radical | C7×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
266 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 80 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)(10,17)(11,19)(12,20)(13,16)(14,18)(15,22)(21,23);(8,10)(9,14)(11,15)(12,21)(13,19)(16,23)(17,20)(18,22);(8,11,12,14,13,17,22,23)(9,10,15,18,16,19,20,21)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5285);