Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#12962 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C13 |
| Frattini, cosocle | C2×#41 | C23×C13 |
| socle | C2×C13 | #1867 |
| prime radical | C13×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
426 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)(14,15,18,22,19,23,30,36)(16,24,26,37,27,38,40,45)(17,21,29,35,28,34,41,44)(20,31,33,39,32,42,43,25);(14,16,18,26,19,27,30,40)(15,20,22,33,23,32,36,43)(17,25,29,31,28,39,41,42)(21,24,35,37,34,38,44,45);(17,28)(25,39)(29,41)(31,42)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5950);