Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C8 |
| direct product | C53×#9835 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | #127 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,21)(16,18)(17,25)(19,24)(20,22)(23,27)(26,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,3,4,2)(5,7,11,6,9,12,8,10)(14,18)(19,26)(20,25)(21,27);(6,10)(7,12)(13,15,22,28,16,23,17,24)(14,19,25,27,18,26,20,21)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2823);