Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #459 |
| 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 ◁ C2×C4×C8×C13 ◁ G | |
294 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)(14,15,17,16)(18,19)(20,30)(21,27)(22,37)(23,40)(24,29)(25,31)(26,35)(28,39)(32,45)(33,47)(34,41)(36,46)(38,42)(43,48)(44,49);(14,16,17,15)(18,20)(19,25)(21,31)(22,38)(23,32)(24,34)(26,41)(27,45)(28,42)(29,44)(30,37)(33,39)(35,48)(36,49)(40,43)(46,47);(18,21,23,35,24,36,39,22)(19,26,28,27,29,37,40,46)(20,31,32,48,34,49,33,38)(25,41,42,45,44,30,43,47)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6168);