Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C4 |
| direct product | C7×#1454 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C23 |
| commutator | C22 | C2×C42×C7 |
| Frattini, cosocle | C42 | C23×C7 |
| socle | C22×C7 | C2×C42 |
| prime radical | C23×C7 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,11,15,12,14,18,21)(10,13,17,19,16,20,22,23)(24,25)(26,29)(27,31)(28,30)(32,36)(33,35)(34,37)(38,39);(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23)(24,26,27,32,28,33,34,38)(25,29,31,36,30,35,37,39);(9,14)(10,16)(15,21)(17,22)(25,30)(29,35)(31,37)(36,39)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4689);