Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62167 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3559 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1684 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)(14,15,18,21)(16,22,23,19)(17,25,26,20)(24,27,29,28)(30,31)(32,39)(33,37)(34,42)(35,38)(36,40)(41,44)(43,45);(14,16)(15,19)(17,24)(18,23)(20,27)(21,22)(25,28)(26,29)(30,31)(32,36)(33,42)(34,37)(35,38)(39,40)(41,45)(43,44);(30,32,35,41)(31,36,38,45)(33,40,34,43)(37,44,42,39);(16,23)(19,22)(24,29)(27,28);(14,17,18,26)(15,20,21,25)(16,24,23,29)(19,27,22,28)(30,33,35,34)(31,37,38,42)(32,40,41,43)(36,44,45,39)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55155);