Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62562 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| 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) = 1664 |
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)(32,41)(33,42)(34,38)(35,39)(36,37)(40,45)(43,44);(14,16)(15,20)(17,24)(18,26)(19,25)(21,28)(22,23)(27,29)(30,32)(31,36)(33,38)(34,43)(35,37)(39,41)(40,44)(42,45);(14,17,19,18)(15,21,22,27)(16,24,25,26)(20,28,23,29)(30,33,34,36,35,44,45,41)(31,37,40,42,39,32,38,43);(16,25)(20,23)(24,26)(28,29)(30,34,35,45)(31,38,39,40)(32,42,37,43)(33,36,44,41);(14,18,19,17)(15,21,22,27)(16,26,25,24)(20,28,23,29)(31,39)(32,37)(38,40)(42,43)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55550);