Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#19012 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C13 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C42×C13 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,20)(17,18)(21,28)(22,25)(23,26)(27,29)(30,31)(32,39)(33,37)(34,42)(35,38)(36,40)(41,44)(43,45);(14,15,19,22)(16,21,24,29)(17,20,18,25)(23,27,26,28)(30,31)(32,36)(33,37)(34,42)(35,38)(39,43)(40,44)(41,45);(14,16)(15,21)(17,23)(18,26)(19,24)(20,27)(22,29)(25,28)(30,32)(31,36)(33,40)(34,43)(35,41)(37,44)(38,45)(39,42);(14,17,19,18)(15,20,22,25)(16,23,24,26)(21,27,29,28)(30,33,35,34)(31,37,38,42)(32,40,41,43)(36,44,45,39)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 12000);