Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C13 · Schur multiplier (H2) = C28×C4 |
| direct product | C4×C13×#519 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | C24 |
| commutator | C2 | C23×C42×C13 |
| Frattini, cosocle | C23 | C25×C13 |
| socle | C23×C13 | C25 |
| prime radical | C24×C13 | C24 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ G | |
3402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4)(25,26,27,28,29,30,31,32,33,34,35,36,37);(5,6,7,8)(10,14)(12,20)(15,17)(18,24);(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26990);