Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C27 |
| direct product | C11×#10980 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C2×C22⋊C4 |
| commutator | C24 | C22×C4×C11 |
| Frattini, cosocle | C25 | C23×C11 |
| prime radical | C22×C11×#121 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C24×C4×C11 ◁ G | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(20,21)(22,26,27,24)(23,25);(13,15)(16,17)(18,19)(20,22)(21,24)(23,27)(25,26);(17,19)(20,22)(21,24)(23,27)(25,26)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 3968);