Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C13 · Schur multiplier (H2) = C28×C4 |
| direct product | C13×#33999 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | C24 |
| commutator | C2 | C23×C42×C13 |
| Frattini, cosocle | C23 | C25×C13 |
| socle | C23×C13 | C25 |
| prime radical | C22×C13×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ G | |
3598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,16)(22,23)(24,30)(25,29)(26,28)(27,31)(32,36)(33,35)(34,37);(18,19,20,21)(22,24)(23,27)(25,32)(26,31)(28,30)(29,34)(33,37)(35,36);(14,16,17,15)(24,31)(27,30)(32,37)(34,36);(23,28)(27,30)(29,35)(34,36);(22,25,26,33)(23,29,28,35)(24,32,31,37)(27,34,30,36)1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26987);