Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#12671 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator | C8 | C2×C42×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C4×D8 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,23,28,24,29,22,27,25,30,39,46,38,45,21,37)(20,31,34,26,35,42,40,43,36,47,32,44,41,49,33,48);(14,15,16,17)(18,20)(19,26)(21,32)(22,34)(23,40)(24,41)(25,36)(27,42)(28,47)(29,48)(30,44)(31,46)(33,39)(35,38)(37,49)(43,45);(18,21,38,39,25,22,24,23)(19,27,45,28,30,37,29,46)(20,32,35,33,36,34,41,40)(26,42,43,47,44,49,48,31)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5659);