Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8×C13 · Schur multiplier (H2) = C24×C4 |
| direct product | C13×#17316 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C13 | C24 |
| commutator | C2 | C22×C4×C8×C13 |
| Frattini, cosocle | C42 | C24×C13 |
| socle | C22×C13 | C22×C42 |
| prime radical | C2×C13×#41 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C13 ◁ G | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(22,23,25,29,26,27,32,34)(24,28,31,35,30,33,36,37);(14,15,16,18,17,19,20,21)(23,27)(28,33)(29,34)(35,37);(22,24)(23,28)(25,31)(26,30)(27,33)(29,35)(32,36)(34,37);(24,30)(28,33)(31,36)(35,37)1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 10304);