Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62591 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C4×C13×#41 ◁ G | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,23)(17,21)(18,27)(19,22)(20,25)(24,29)(26,28)(30,31,35,38)(32,36,34,43)(33,40,42,44)(37,39,45,41);(14,16)(15,20)(17,24)(18,26)(19,25)(21,28)(22,23)(27,29)(30,32,35,34)(31,36,38,43)(33,41,42,39)(37,44,45,40);(14,17,19,18)(15,21,22,27)(16,24,25,26)(20,28,23,29)(30,33,34,41,35,42,32,39)(31,37,36,40,38,45,43,44);(16,25)(20,23)(24,26)(28,29)(33,42)(37,45)(39,41)(40,44);(14,18,19,17)(15,21,22,27)(16,26,25,24)(20,28,23,29)(33,42)(37,45)(39,41)(40,44)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55579);