Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#10608 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C23×C53 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,5,6,4,8,3,7)(9,10,15,18)(11,14,17,13)(12,20,23,19)(16,21,24,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(9,11,13,10,15,17,14,18)(12,20,21,16,23,19,22,24);(2,6)(3,5)(7,8)(9,12,15,23)(10,16,18,24)(11,19,17,20)(13,22,14,21)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3596);