Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#27069 |
| 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 | |
734 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)(14,15,18,19)(16,20,17,21)(22,23,25,26)(24,28,29,27)(30,31,33,36)(32,38,40,35)(34,37,42,45)(39,44,41,43);(15,19)(20,21)(23,26)(27,28)(30,32,33,40)(31,35,36,38)(34,39,42,41)(37,43,45,44);(14,16,18,17)(15,20,19,21)(22,24,25,29)(23,27,26,28)(32,39)(34,42)(35,43)(37,45)(38,44)(40,41);(14,17,18,16)(15,20,19,21)(23,26)(27,28)(31,36)(35,38)(37,45)(43,44)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 20057);