Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#26791 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×#121×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C4×C22⋊C4×C53 ◁ G | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 81 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,26)(24,28)(25,27)(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);(1,2,4,3)(13,15)(14,18)(16,23)(17,22)(19,21)(20,28)(24,25)(26,27);(1,2,4,3)(6,9)(10,12)(14,19)(15,22)(17,25)(18,26)(21,28)(23,24);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,19)(18,21)(20,27)(26,28)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 19779);