Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#22612 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator | C2×C4 | C23×C4×C13 |
| Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C24×C13 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C4×C13×C22⋊C4 ◁ G | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17)(18,19)(20,22)(21,23)(24,25)(26,27,31,32,30,33,29,35)(28,34,38,39,37,40,36,41);(18,20,21,24)(19,22,23,25)(27,32)(28,36)(29,31)(33,35)(37,38)(39,41);(19,23)(22,25)(27,33)(32,35)(34,40)(39,41);(26,28,31,38,30,37,29,36)(27,34,32,39,33,40,35,41)910 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 15600);