Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#32822 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C22×D4×C13 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C43×C13 ◁ G | |
626 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)(14,15,17,19)(16,18,20,21)(22,23)(24,29)(25,27)(26,28)(30,31,34,38)(32,40,42,36)(33,37,35,39)(41,45,43,44);(14,16)(15,18)(17,20)(19,21)(22,24,26,25)(23,27,28,29)(32,41)(33,35)(36,44)(37,39)(40,45)(42,43);(15,19)(18,21)(23,28)(24,25)(30,32)(31,36)(33,41)(34,42)(35,43)(37,44)(38,40)(39,45);(16,20)(18,21)(22,25,26,24)(23,27,28,29)(30,33,34,35)(31,37,38,39)(32,41,42,43)(36,44,40,45)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 25810);