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×#12665 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C2×D4 |
| commutator | C4 | C2×C4×C8×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C8×D4 |
| prime radical | C23×C13 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C22×C13×C16 ◁ G | |
326 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)(18,19,23,27,22,26,35,21,24,28,40,36,38,45,49,37)(20,29,32,30,33,25,41,42,34,43,39,48,31,44,47,46);(14,15,16,17)(18,20)(19,25)(21,30)(22,33)(23,39)(24,34)(26,43)(27,46)(28,44)(29,45)(31,38)(32,40)(35,47)(36,42)(37,48)(41,49);(18,21,35,45,38,27,23,28,24,37,49,26,22,36,40,19)(20,30,47,29,31,46,39,44,34,48,41,43,33,42,32,25)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5653);