Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#24722 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×#41×C53 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,4,5)(3,7,8,6)(9,10,14,16,13,17,12,18)(11,19,23,20,22,15,21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(2,5)(6,7)(9,11)(10,15)(12,21)(13,22)(14,23)(16,24)(17,19)(18,20);(9,10,13,17)(11,20,22,24)(12,16,14,18)(15,23,19,21);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,16,17,18)(11,21,22,23)(15,24,19,20)2279 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 17710);