Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#2102 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #400 |
| prime radical | C23×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
158 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 158 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)(8,9,12,11,13,15,17,21)(10,16,18,23,19,20,22,14);(8,10,17,18,13,19,12,22)(9,14,21,16,15,23,11,20);(8,11,12,15,13,21,17,9)(10,16,18,23,19,20,22,14)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5337);