Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#13201 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #390 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1955 |
| prime radical | C13×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C8×C13 ◁ G | |
420 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)(14,15,19,25,20,24,39,43,21,26,40,29,18,23,28,34)(16,27,31,45,30,37,42,44,32,35,22,41,38,17,33,36);(14,16,28,42,18,38,40,31,21,32,39,33,20,30,19,22)(15,17,34,45,23,35,29,36,26,37,43,41,24,27,25,44);(14,17,18,35,21,37,20,27)(15,22,24,33,26,31,23,42)(16,29,30,34,32,25,38,43)(19,36,28,41,40,44,39,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6189);