Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11351 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #381 |
| commutator | C24 | C22×C4×C53 |
| Frattini, cosocle | C25 | C23×C53 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C25×C53 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 73 points:
(5,6)(7,10,14,18)(8,15)(9,12)(11,16)(13,17,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(1,2)(3,4)(5,7)(6,10,12,18)(8,13)(9,14)(11,17,15,20)(16,19);(2,4)(5,8)(6,11,12,15)(7,13)(9,16)(10,17,18,20)(14,19)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4339);