Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#12757 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #1790 |
| commutator | C2×D4 | C22×C4×C13 |
| Frattini, cosocle | C2×#41 | C23×C13 |
| prime radical | C13×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
736 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)(14,15,18,22,19,23,32,39)(16,24,27,41,26,40,44,45)(17,28,30,38,29,21,31,36)(20,33,34,25,35,42,37,43);(14,16,19,26)(15,20,23,35)(17,25,29,43)(18,27,32,44)(21,24,28,40)(22,37,39,34)(30,42,31,33)(36,45,38,41);(21,28)(24,40)(36,38)(41,45)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5745);