Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#13017 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #447 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C22×C13×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19,21,26,36,45,22,38,24,29,37,46,23,28,35,27)(20,30,31,47,40,42,49,44,34,48,39,43,33,41,32,25);(14,15,16,17)(18,20)(19,25)(21,32)(22,39)(23,40)(24,34)(26,41)(27,30)(28,47)(29,44)(31,35)(33,36)(37,49)(38,48)(42,46)(43,45);(19,26)(21,35)(22,37)(23,36)(25,41)(27,45)(28,38)(29,46)(30,43)(31,32)(33,40)(39,49)(42,44)(47,48)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6005);