Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#19150 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C13 |
| socle | C22×C13 | C2×#121 |
| prime radical | C23×C13 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ G | |
586 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)(16,23)(17,21)(18,27)(19,22)(20,26)(24,29)(25,28)(30,31,34,37)(32,38,40,45)(33,41,43,36)(35,42,44,39);(14,16,18,26,19,24,17,25)(15,20,21,23,22,28,27,29);(16,24)(20,28)(23,29)(25,26)(30,32,34,40)(31,35,37,44)(33,42,43,39)(36,45,41,38);(14,17,19,18)(15,21,22,27)(16,25,24,26)(20,23,28,29)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 12138);