Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#2126 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #99 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×D4 | C23×C7 |
| socle | C2×C7 | C2×#99 |
| prime radical | C7×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
258 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 84 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)(10,14)(11,15,19,13)(12,16)(17,21,23,20)(18,22);(9,13)(11,19)(14,20)(15,16)(17,23)(21,22);(8,10,12,18)(9,14,16,22)(11,17,19,23)(13,20,15,21)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5361);