Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13127 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #388 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1924 |
| prime radical | C2×C13×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C8×C13 ◁ G | |
300 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)(14,15,19,25,18,23,27,22,20,24,36,42,34,43,17,32)(16,26,28,41,30,21,33,39,31,40,35,44,29,38,45,37);(14,16,27,45,34,29,19,35,20,31,17,33,18,30,36,28)(15,21,22,41,43,26,25,37,24,38,32,44,23,40,42,39);(14,17,18,19,20,27,34,36)(15,22,23,42,24,32,43,25)(16,28,29,45,31,35,30,33)(21,37,26,44,38,39,40,41)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6115);