Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#23820 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| socle, prime radical | C23×C53 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C43×C53 ◁ G | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,20,23)(19,25,27,22)(21,24,28,31)(26,32,29,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,4,8)(2,5,6,7)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(2,6)(3,8)(10,13)(14,16)(17,19,20,27)(18,22,23,25)(21,29,28,26)(24,32,31,30);(9,11,12,15)(10,14,13,16)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16808);