Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#13115 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #481 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1919 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C13×#97 ◁ G | |
252 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)(14,15,20,25,18,24,21,35)(16,26,29,43,28,22,19,33)(17,30,31,37,32,45,36,42)(23,38,39,44,41,34,40,27);(14,16,20,29,18,28,21,19)(15,22,25,33,24,26,35,43)(17,27,31,38,32,44,36,34)(23,37,39,45,41,42,40,30);(14,17,20,31,18,32,21,36)(15,23,35,40,24,41,25,39)(16,27,29,38,28,44,19,34)(22,37,43,30,26,42,33,45)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6103);