Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11291 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C2×D4 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C23×C53 | C4×D4 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C23×C8×C53 ◁ G | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 77 points:
(6,7)(10,15,18,17)(16,23,24,19)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,2)(3,4)(5,6,8,7)(9,10)(11,19)(12,17)(13,15)(14,18)(16,21)(20,24)(22,23);(2,4)(5,7,8,6)(9,11,12,20,14,22,13,21)(10,16,15,23,18,24,17,19)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4279);