Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13281 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 3328 |
A minimal faithful permutation representation — 3 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17)(18,19)(20,24)(21,22)(23,25)(26,27)(28,38)(29,35)(30,45)(31,47)(32,37)(33,40)(34,43)(36,46)(39,52)(41,48)(42,54)(44,51)(49,56)(50,55)(53,57);(15,17)(19,22)(23,24)(26,28,29,39,31,41,43,55,32,42,44,56,46,57,30,40)(27,33,45,53,36,49,51,54,37,50,34,48,47,52,35,38);(18,20,21,25)(19,23,22,24)(26,29,31,43,32,44,46,30)(27,34,36,35,37,45,47,51)(28,39,41,55,42,56,57,40)(33,48,49,38,50,53,52,54)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6269);