Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C53 · Schur multiplier (H2) = C213 |
| direct product | C53×#62788 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C53 |
| socle | C22×C53 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
7094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 6 generators on 77 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(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);(2,5)(6,8);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23);(10,16)(14,17)(15,24)(22,23);(1,3,4,7)(2,6,5,8)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 55776);