Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #481 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C2×C13×#97 ◁ G | |
318 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)(15,16)(18,19)(20,28,31,44)(21,24)(22,34)(23,37)(25,30,39,46)(26,36)(27,29)(32,47,38,42)(33,48)(35,41,49,43)(40,45);(14,15,17,16)(19,24)(22,33)(25,39)(27,37)(30,46)(35,49)(36,45)(41,43);(14,16,17,15)(18,20,29,32,21,31,23,38)(19,25,37,41,24,39,27,43)(22,35,45,46,33,49,36,30)(26,42,48,44,40,47,34,28)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6127);