Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13190 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C53 | C23 |
| commutator | C23 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C23×C53 | C22×C8 |
| prime radical | C22×D4×C53 | C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C53 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,4)(6,8)(9,10,13,17,14,15,21,22,12,16,20,23,11,18,19,24)(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);(2,4)(9,10,13,17,14,15,21,22,12,16,20,23,11,18,19,24);(5,6,7,8)(9,11,12,14)(10,15,16,18)(13,19,20,21)(17,22,23,24)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6178);