Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#21292 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C13 | C2×D4 |
| commutator | C2×C4 | C23×C4×C13 |
| Frattini, cosocle | C22×C4 | C24×C13 |
| socle | C22×C13 | C2×C4×D4 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,19)(17,18)(20,21)(22,23,27,31,28,32,26,36)(24,29,33,37,34,30,25,35);(15,18)(19,21)(22,24,27,33,28,34,26,25)(23,29,31,37,32,30,36,35);(22,25,26,34,28,33,27,24)(23,30,31,35,32,29,36,37);(14,16,17,20)(15,19,18,21)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 14280);