Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C24 |
| direct product | C2×C7×#339 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | D4 |
| commutator | C4 | C2×C42×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C23×C7 | C22⋊C4 |
| prime radical | C24×C7 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C23×C4×C7 ◁ G | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(15,16,17,18,19,20,21);(3,4,5,6)(7,8)(9,11)(10,13)(12,14);(8,11,13,12)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4967);