Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13271 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #476 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1987 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
332 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)(14,15,18,24,20,25,34,44,21,26,33,23,19,36,17,32)(16,27,28,39,30,43,45,22,31,40,29,41,37,38,35,42);(14,16)(15,22,26,42)(17,28)(18,35)(19,30)(20,37)(21,31)(23,38,24,43)(25,39,36,41)(27,32,40,44)(29,34)(33,45);(14,17,19,33,21,34,20,18)(15,23,25,32,26,24,36,44)(16,28,30,45,31,29,37,35)(22,38,39,40,42,43,41,27)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6259);