Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#2931 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C23 |
| commutator | C22 | C23×C4×C7 |
| Frattini, cosocle | C2×C4 | C24×C7 |
| socle | C22×C7 | C23×C4 |
| prime radical | C7×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,20,16,19,25,27)(14,21,23,24,22,18,26,15);(9,11)(12,14)(13,18)(15,20)(16,22)(17,23)(19,21)(24,27)(25,26);(12,15,17,21,16,24,25,18)(13,14,20,23,19,22,27,26);(13,19)(14,22)(20,27)(23,26)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17006);