Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C16 · Schur multiplier (H2) = C1 |
| direct product | C7×#1422 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C16 | D4 |
| commutator | C4 | C2×C7×C16 |
| Frattini, cosocle | C2×C16 | C22×C7 |
| socle | C2×C7 | C4⋊C16 |
| prime radical | C22×C7 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C16 ◁ C2×C7×C16 ◁ G | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,18,13,19,27,34,14,17,28,32,24,33,37,39,15,20,29,35,22,16,30,31,11,25,26,38,10,21,23,36);(8,10,11,22,15,24,14,13)(9,16,17,21,20,19,25,33)(12,23,26,30,29,37,28,27)(18,31,32,36,35,34,38,39)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 554);