Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#19338 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
636 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)(3,5)(4,6)(7,8)(9,10,14,12)(11,17,20,24)(13,21,22,16)(15,23,18,19)(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)(9,11,13,19,14,20,22,23)(10,15,16,17,12,18,21,24);(2,6)(4,8)(5,7)(10,12)(15,18)(16,21)(17,24);(2,5)(6,7)(9,12,14,10)(11,18,20,15)(13,21,22,16)(17,19,24,23)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12326);