Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62582 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3514 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C4×C13×#41 ◁ G | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 832 |
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)(32,39)(33,37)(34,43)(35,38)(36,40)(41,44)(42,45);(14,16)(15,20)(17,24)(18,26)(19,25)(21,28)(22,23)(27,29);(14,17,19,18)(15,21,22,27)(16,24,25,26)(20,28,23,29)(30,32,34,40,35,41,33,42)(31,36,37,39,38,45,43,44);(16,25)(20,23)(24,26)(28,29);(14,18,19,17)(15,21,22,27)(16,26,25,24)(20,28,23,29)(30,33,35,34)(31,37,38,43)(32,40,41,42)(36,44,45,39)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55570);