Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#7924 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | C2×#95 |
| prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ C22×C42×C53 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,4,7,6,8,5,3)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(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);(1,3,5,8,6,7,4,2)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 912);