Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#1342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #327 |
| commutator | C2×C8 | C2×C4×C7 |
| Frattini, cosocle | C2×M16 | C22×C7 |
| prime radical | C23×C7 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×M16 ◁ G | |
100 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)(8,9,12,17,14,18,13,29)(10,19,21,33,23,36,22,39)(11,24,26,16,28,34,27,35)(15,25,30,38,32,37,31,20);(8,10,14,23)(9,15,18,32)(11,25,28,37)(12,22,13,21)(16,33,35,39)(17,30,29,31)(19,34,36,24)(20,27,38,26)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 474);