Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62146 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C4×C13×#41 ◁ G | |
1700 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)(32,40)(33,39)(34,36)(35,38)(37,41)(42,45)(43,44);(14,16)(15,19)(17,24)(18,23)(20,27)(21,22)(25,28)(26,29)(31,36)(37,44)(38,39)(40,45);(30,32,34,42)(31,37,36,44)(33,41,35,43)(38,40,39,45);(16,23)(19,22)(24,29)(27,28)(31,36)(37,44)(38,39)(40,45);(14,17,18,26)(15,20,21,25)(16,24,23,29)(19,27,22,28)(30,33,34,35)(31,38,36,39)(32,41,42,43)(37,40,44,45)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55134);