Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#2896 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D4 |
| commutator | C4 | C23×C4×C7 |
| Frattini, cosocle | C2×C4 | C24×C7 |
| socle | C22×C7 | C22×D4 |
| prime radical | C22×C7×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,21)(15,20)(16,24)(17,19)(18,22)(23,27)(25,26);(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(13,19)(18,26)(20,24)(21,27);(12,15,17,16)(13,20,19,24)(14,22,23,25)(18,27,26,21)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 16971);