Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#18016 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22×D4 |
| commutator | C2×C4 | C23×C4×C13 |
| Frattini, cosocle | C42 | C24×C13 |
| socle | C22×C13 | C2×C4×D4 |
| prime radical | C2×C13×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C13×#431 ◁ G | |
600 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)(14,15,19,23)(16,21,25,28)(17,22,18,27)(20,26,29,24)(30,31,33,36,34,37,40,43)(32,35,39,42,38,41,44,45);(14,16)(15,20)(17,24)(18,26)(19,25)(21,27)(22,28)(23,29);(14,16)(15,21)(17,26)(18,24)(19,25)(20,27)(22,29)(23,28)(30,32,33,39,34,38,40,44)(31,35,36,42,37,41,43,45);(14,17,19,18)(15,22,23,27)(16,24,25,26)(20,28,29,21)(32,38)(35,41)(39,44)(42,45)910 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 11004);