Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2161 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×D8 |
| commutator | C8 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C2×#107 |
| prime radical | C23×C7 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
140 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)(8,9,12,17,14,18,13,28)(10,19,23,35,21,33,24,38)(11,25,26,32,27,39,20,16)(15,29,31,37,30,36,34,22);(8,10,20,36,13,24,27,37,14,21,26,29,12,23,11,22)(9,15,25,35,17,31,32,33,18,30,39,38,28,34,16,19);(10,21)(15,30)(19,33)(22,37)(23,24)(29,36)(31,34)(35,38)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5396);