Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C27×C4 |
| direct product | C22×C53×#491 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ C22×C42×C53 ◁ G | |
2142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 5 generators on 81 points:
(1,2)(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);(3,4);(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,21,26,27)(16,20,25,28)(17,23,18,22);(5,7,8,11)(6,9,10,12)(13,15,17,16,19,26,18,25)(14,20,22,21,24,28,23,27);(6,10)(9,12)(13,16,17,26,19,25,18,15)(14,21,22,28,24,27,23,20)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53596);