Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#26335 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C4×Q8×C13 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 3328 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,19)(16,20,17,21)(22,23)(24,27)(25,29)(26,28)(30,31,35,39)(32,36,42,45)(33,37,34,38)(40,43,41,44);(15,19)(20,21)(22,24,26,25)(23,27,28,29)(30,32,35,42)(31,36,39,45)(33,41,34,40)(37,44,38,43);(14,16,18,17)(15,20,19,21)(22,25,26,24)(23,27,28,29)(32,40)(33,34)(36,43)(37,38)(41,42)(44,45);(14,17,18,16)(15,20,19,21)(30,33,35,34)(31,37,39,38)(32,41,42,40)(36,44,45,43)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 19323);