Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#29031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C22×D4×C13 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C4×C13×C4⋊C4 ◁ G | |
674 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)(14,15,17,19)(16,20,21,18)(22,23,25,26)(24,28,29,27)(30,31)(32,35)(33,37)(34,36)(38,42)(39,44)(40,43)(41,45);(14,16,17,21)(15,18,19,20)(23,26)(27,28)(30,32,33,40,34,39,41,38)(31,35,37,43,36,44,45,42);(23,26)(27,28)(32,38)(33,41)(35,42)(37,45)(39,40)(43,44);(15,19)(18,20)(22,24,25,29)(23,27,26,28)(31,36)(32,39)(37,45)(38,40)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22019);