Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#12002 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | #106 |
| commutator | C8 | C22×C8×C13 |
| Frattini, cosocle | C4×C8 | C23×C13 |
| socle | C2×C13 | C2×#334 |
| prime radical | C23×C13 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
220 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)(14,15,18,24,20,25,35,41,21,26,34,23,19,37,17,33)(16,27,30,39,32,42,29,38,28,40,36,45,31,44,43,22);(14,16,17,29,19,31,34,30,21,28,35,43,20,32,18,36)(15,22,23,27,25,39,33,42,26,38,24,40,37,45,41,44);(16,28)(22,38)(27,40)(29,43)(30,36)(31,32)(39,45)(42,44)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 4990);