Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#23801 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| socle, prime radical | C23×C53 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C43×C53 ◁ G | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 81 points:
(5,6,8,9)(7,10,11,12)(13,14,17,23,19,25,18,24)(15,26,27,21,16,22,28,20)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,2)(3,4)(6,9)(10,12)(13,15,19,16)(14,20,25,21)(17,28,18,27)(22,24,26,23);(2,4)(13,15,19,16)(14,21,25,20)(17,28,18,27)(22,23,26,24);(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,22,25,26)(17,28,18,27)(20,24,21,23)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16789);