Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13188 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #476 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1948 |
| prime radical | C23×C13 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,19,26,21,27,20,25)(16,28,32,42,33,39,31,45)(17,34,35,23,36,24,18,37)(22,38,40,30,41,44,43,29);(14,16)(15,22,27,41)(17,29)(18,38)(19,31)(20,32)(21,33)(23,39,37,28)(24,42,34,45)(25,40,26,43)(30,36)(35,44);(14,17,19,35,21,36,20,18)(15,23,25,34,27,37,26,24)(16,29,31,44,33,30,32,38)(22,39,40,45,41,28,43,42)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6176);