Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#29133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C2×C13×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C13×#376 ◁ G | |
754 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)(22,23,25,27)(24,26,28,29)(30,31,33,37,34,38,41,36)(32,35,40,44,39,43,45,42);(15,18)(19,20)(22,24,25,28)(23,26,27,29)(30,32)(31,35)(33,40)(34,39)(36,42)(37,44)(38,43)(41,45);(24,28)(26,29)(31,36)(33,41)(35,42)(37,38)(40,45)(43,44);(14,16,17,21)(15,19,18,20)(32,39)(35,43)(40,45)(42,44)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22121);