Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×C53×#2163 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C53 | C23 |
| commutator | C22 | C23×C8×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C23×C53 | C22×C8 |
| prime radical | C24×C53 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C53 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 79 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55);(56,57)(58,61)(59,60)(62,63)(64,65,66,70,67,71,72,77,68,69,73,75,74,76,78,79);(57,60)(61,63);(56,58,59,62)(57,61,60,63)(65,69)(70,75)(71,76)(77,79)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26404);