Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#29424 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C2×C13×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C13×#117 ◁ G | |
780 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)(16,20)(17,18)(19,21)(22,23)(24,28)(25,27)(26,29)(30,31,33,37)(32,39,40,35)(34,36,42,38)(41,44,45,43);(15,18)(19,20)(22,24)(23,26)(25,29)(27,28)(30,32,33,40)(31,35,37,39)(34,41,42,45)(36,43,38,44);(24,29)(26,28)(31,36)(34,42)(35,43)(37,38)(39,44)(41,45);(14,16,17,21)(15,19,18,20)(23,27)(26,28)(31,37)(32,40)(36,38)(41,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22412);