Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C11 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C11×#3221 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C11 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C2×C11×#117 ◁ G | |
2150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37);(3,4)(5,9)(6,8)(7,10)(13,19)(14,21)(16,23)(18,25)(20,22)(24,26);(3,5,6,10)(4,7,8,9)(11,12)(13,16)(14,18)(15,17)(19,23)(20,24)(21,25)(22,26);(5,10)(7,9)(11,13,15,20)(12,16,17,24)(14,22,21,19)(18,26,25,23);(4,8)(7,9)(12,17)(16,24)(18,25)(23,26)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 53782);