Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#26895 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C4×C4⋊C4×C53 ◁ G | |
788 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)(7,11,12,10)(13,14,17,22,19,24,18,23)(15,25,26,21,16,28,27,20)(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)(10,11)(13,15,19,16)(14,20,24,21)(17,27,18,26)(22,28,23,25);(2,4)(13,15,19,16)(14,21,24,20)(17,27,18,26)(22,25,23,28);(5,7,8,12)(6,10,9,11)(13,16,19,15)(14,21,24,20)(17,26,18,27)(22,25,23,28)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 19883);