Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#12678 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #442 |
| commutator | C8 | C2×C42×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1776 |
| prime radical | C23×C13 | #137 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C13×#97 ◁ G | |
292 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)(14,15,18,25,20,26,34,24,21,27,35,41,19,23,17,33)(16,28,30,39,31,40,44,43,32,42,45,22,36,38,29,37);(14,16,19,31,21,32,20,36)(15,22,23,37,27,39,26,43)(17,30,35,44,34,45,18,29)(24,40,25,42,33,38,41,28);(15,23,27,26)(17,34)(18,35)(22,37,39,43)(24,41,33,25)(28,38,42,40)(29,44)(30,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5666);