Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#12941 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #447 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,16)(19,25)(20,30)(21,22)(23,35)(26,42)(27,28)(29,41)(31,32)(33,49)(34,38)(36,37)(39,48)(40,44)(45,46);(14,16,17,15)(18,19)(20,26)(21,34)(22,25)(23,38)(24,29)(27,36)(28,35)(30,43)(31,40)(32,42)(33,46)(37,41)(39,45)(44,48)(47,49);(18,20,24,33)(19,26,29,46)(21,30,36,49)(22,31,37,48)(23,39,35,32)(25,40,41,44)(27,47,34,43)(28,42,38,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5929);