Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43×C11 · Schur multiplier (H2) = C23×C43 |
| direct product | C42×C11×C4⋊C4 |
| unambiguous semidirect product | C4⋊(C43×C11) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42×C11 | C22 |
| commutator | C2 | C2×C43×C11 |
| Frattini, cosocle | C24 | C24×C11 |
| socle, prime radical | C24×C11 | C24 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42×C11 ◁ G | |
2478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 2816 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4)(17,18,19,20,21,22,23,24,25,26,27);(5,6,7,8);(10,12)(13,14,15,16);(9,10,11,12)1760 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6736);