Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C13 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×C13×#405 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8×C13 | C22 |
| commutator | C2 | C23×C42×C13 |
| Frattini, cosocle | C2×C4 | C25×C13 |
| socle | C24×C13 | C22×C4 |
| prime radical | C25×C13 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8×C13 ◁ G | |
3518 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)(25,26,27,28,29,30,31,32,33,34,35,36,37);(3,4);(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(5,6,7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)2080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 53041);