Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2110 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | C42 |
| commutator | C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C2×#97 |
| prime radical | C23×C7 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ G | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,11,15,12,17,23,31,13,16,24,30,25,32,38,39)(10,18,20,34,21,27,36,29,19,33,22,37,35,14,26,28);(8,10)(9,14,16,27)(11,22)(12,21)(13,19)(15,29,30,28)(17,18,32,33)(20,24)(23,26)(25,35)(31,37,39,34)(36,38);(10,19)(14,27)(18,33)(20,22)(21,35)(26,36)(28,29)(34,37)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5345);