Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#7181 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | #102 |
| commutator | C8 | C4×C8×C13 |
| Frattini, cosocle | C2×C4×C8 | C22×C13 |
| socle | C2×C13 | #1279 |
| prime radical | C23×C13 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,23,20,25,36,41,21,26,35,42,19,24,17,33)(16,27,29,38,31,44,28,22,32,37,45,39,30,43,34,40);(14,16,19,30,21,32,20,31)(15,22)(17,34,35,45,36,28,18,29)(23,37)(24,38)(25,39)(26,40)(27,42)(33,44)(41,43)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 169);