Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1914 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #386 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C7×#535 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
340 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 108 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)(8,9,12,16,13,15,22,10)(11,18,20,23,19,14,21,17);(8,10,12,15,13,16,22,9)(11,17,21,18,19,23,20,14);(8,11)(9,14)(10,17)(12,21)(13,19)(15,18)(16,23)(20,22)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5149);