Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62575 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3546 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,23)(17,21)(18,27)(19,22)(20,25)(24,29)(26,28)(30,31,34,37)(32,39,41,45)(33,36,43,38)(35,40,44,42);(14,16)(15,20)(17,24)(18,26)(19,25)(21,28)(22,23)(27,29)(30,31)(32,35)(33,38)(34,37)(36,43)(39,42)(40,45)(41,44);(14,17,19,18)(15,21,22,27)(16,24,25,26)(20,28,23,29)(30,32,33,42,34,41,43,40)(31,35,38,39,37,44,36,45);(16,25)(20,23)(24,26)(28,29)(30,33,34,43)(31,36,37,38)(32,40,41,42)(35,39,44,45);(14,18,19,17)(15,21,22,27)(16,26,25,24)(20,28,23,29)(31,37)(35,44)(36,38)(39,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55563);