Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C53 · Schur multiplier (H2) = C42 |
| direct product | C53×#12594 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C42 |
| commutator | C4 | C2×C4×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C2×#97 |
| prime radical | C23×C53 | C2×M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,5,4,6,7,8)(9,10,13,19,14,16,11,17,15,20,22,23,12,18,21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(10,16,20,18)(11,21)(13,22)(17,19,24,23);(9,11,12,13,15,21,14,22)(10,17,18,19,20,24,16,23)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5582);