Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#12782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #1790 |
| commutator | C2×Q8 | C22×C4×C13 |
| Frattini, cosocle | C22×Q8 | C23×C13 |
| prime radical | C13×#3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C23×C13 ◁ C22×Q8×C13 ◁ G | |
730 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)(14,15,20,26,21,27,19,25)(16,28,30,44,31,42,36,38)(17,32,34,23,18,24,33,41)(22,37,39,29,40,45,43,35);(14,16)(15,22)(17,29)(18,35)(19,36)(20,30)(21,31)(23,38)(24,42)(25,39)(26,43)(27,40)(28,32)(33,37)(34,45)(41,44);(14,17,21,18)(15,23,27,41)(16,29,31,35)(19,34,20,33)(22,38,40,44)(24,26,32,25)(28,39,42,43)(30,37,36,45)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5770);