Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#33359 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C22×#99 |
| commutator | C2×C4 | C23×C4×C11 |
| Frattini, cosocle | C2×D4 | C24×C11 |
| prime radical | C11×#3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×D4×C11 ◁ C2×C11×H(5,𝔽2) ◁ G | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,18)(15,19,23,17)(16,20)(21,25)(22,26,27,24);(13,17)(15,23)(18,24)(19,20)(22,27)(25,26);(12,14)(13,18)(15,22)(16,21)(17,24)(19,26)(20,25)(23,27);(14,21)(18,25)(22,27)(24,26)506 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 26347);