Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61861 |
| 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×#533 ◁ G | |
1670 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)(16,22)(17,20)(18,21)(19,24)(23,28)(25,27)(26,29)(31,36)(32,39)(33,35)(37,44)(40,42)(43,45);(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,28,29,27)(30,31,34,36)(32,37,40,43)(33,41,35,38)(39,45,42,44);(30,32)(31,37)(33,39)(34,40)(35,42)(36,43)(38,44)(41,45);(14,17,18,26)(15,20,21,29)(16,23,24,25)(19,27,22,28)(31,36)(37,43)(38,41)(44,45);(15,21)(19,22)(20,29)(27,28)(30,33,34,35)(31,38,36,41)(32,39,40,42)(37,44,43,45)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54849);