Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#23662 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×#117×C53 ◁ G | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(15,22)(19,26)(23,28)(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);(5,7,9,8)(6,10,11,12)(13,14)(15,19)(16,18)(17,21)(20,24)(22,26)(23,25)(27,28);(2,4)(5,8,9,7)(6,10,11,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(13,15,17,22)(14,19,21,26)(16,23,24,28)(18,25,20,27)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16650);