Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C13 · Schur multiplier (H2) = C29 |
| direct product | C13×#34199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C25 |
| commutator | C22 | C24×C4×C13 |
| Frattini, cosocle | C23 | C25×C13 |
| prime radical | C23×D4×C13 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ G | |
3190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19,21,20)(22,23)(24,30)(25,33)(26,27)(28,32)(29,34)(31,35)(36,37);(18,20,21,19)(23,27)(28,30)(29,33)(35,36);(14,15)(16,17)(22,24,26,32)(23,28,27,30)(25,31,34,37)(29,35,33,36);(22,23)(24,28)(25,29)(26,27)(30,32)(31,35)(33,34)(36,37);(22,25)(23,29)(24,31)(26,34)(27,33)(28,35)(30,36)(32,37)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 27187);