Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C32 · Schur multiplier (H2) = C22 |
| direct product | C7×C22⋊C32 |
| unambiguous semidirect product | C22⋊(C7×C32) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C16 | C22 |
| commutator | C2 | C2×C7×C32 |
| Frattini, cosocle | C2×C16 | C22×C7 |
| socle | C22×C7 | C2×C16 |
| prime radical | C23×C7 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C16 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,14,18,15,19,22,28,16,20,23,29,25,31,34,38,17,21,24,30,26,32,35,39,27,33,36,40,37,41,42,43);(9,11)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 531);