Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C7×#1327 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #99 |
| commutator | C2×C4 | C2×C7×C8 |
| Frattini, cosocle | C2×M16 | C22×C7 |
| socle | C2×C7 | #324 |
| prime radical | C23×C7 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×M16 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7)(8,9,11,16,12,15,20,22,13,17,19,23,10,14,18,21);(9,14,17,15)(11,19)(16,21,23,22)(18,20)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 459);