Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#32733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C24×C13 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C13×#117 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,20)(16,19,21,18)(22,23)(24,26)(25,27)(28,29)(30,31,34,33)(32,38,40,36)(35,37,43,42)(39,41,45,44);(14,16)(15,18)(17,21)(19,20)(22,24)(23,26)(25,28)(27,29)(30,32,34,40)(31,36,33,38)(35,44,43,41)(37,45,42,39);(14,16)(15,19)(17,21)(18,20)(23,27)(26,29)(30,33,34,31)(32,39,40,45)(35,37,43,42)(36,44,38,41);(16,21)(18,19)(24,28)(26,29)(31,33)(32,40)(37,42)(41,44)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 25721);