Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#27052 |
| 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 ◁ C4×C13×C22⋊C4 ◁ G | |
694 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)(17,21)(18,19)(22,23)(24,28)(25,26)(27,29)(30,31)(32,39)(33,43)(34,38)(35,37)(36,42)(40,45)(41,44);(15,19)(20,21)(23,26)(27,28)(30,32,34,42)(31,36,38,39)(33,41,35,40)(37,44,43,45);(14,16,18,17)(15,20,19,21)(22,24,25,29)(23,27,26,28)(32,40)(33,35)(36,44)(37,43)(39,45)(41,42);(14,17,18,16)(15,20,19,21)(23,26)(27,28)(30,33,34,35)(31,37,38,43)(32,41,42,40)(36,44,39,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 20040);