Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#12656 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C2×D4 |
| commutator | C4 | C2×C4×C8×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C2×C4⋊C8 |
| prime radical | C23×C13 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
358 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)(20,32,35,45)(21,37)(22,28)(23,29)(24,30)(25,31)(26,34,42,40)(27,39)(33,43,47,46)(36,41,48,49)(38,44);(18,20,23,36,24,34,39,47,25,35,38,48,22,40,21,33)(19,26,29,43,30,32,27,41,31,42,44,46,28,45,37,49);(14,16,17,15)(18,21,22,38,25,39,24,23)(19,27,28,29,31,37,30,44)(20,33,40,48,35,47,34,36)(26,41,45,43,42,49,32,46)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5644);