Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#26839 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C22×D4×C53 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C4×C4⋊C4×C53 ◁ G | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 81 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,20,25,28)(16,26,24,19)(17,27,23,21)(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)(13,15,23,16,18,25,17,24)(14,19,27,28,22,26,21,20);(2,4)(6,9)(10,12)(13,15,17,24,18,25,23,16)(14,19,21,20,22,26,27,28);(5,7,8,11)(6,10,9,12)(13,16,23,25,18,24,17,15)(14,20,27,19,22,28,21,26)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 19827);