Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#32783 |
| 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 | |
686 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)(16,18)(17,19)(20,21)(22,23,26,25)(24,28,29,27)(30,31,34,38)(32,39,41,36)(33,42,35,37)(40,45,43,44);(14,16)(15,18)(17,20)(19,21)(22,24,26,29)(23,27,25,28)(32,40)(33,35)(36,44)(37,42)(39,45)(41,43);(15,19)(18,21)(23,25)(27,28)(30,32)(31,36)(33,40)(34,41)(35,43)(37,45)(38,39)(42,44);(16,20)(18,21)(22,25,26,23)(24,28,29,27)(30,33,34,35)(31,37,38,42)(32,40,41,43)(36,45,39,44)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 25771);