Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C53×#25926 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×AGL(1,ℤ/8)×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,16)(12,22)(13,17)(14,19)(15,20)(18,23)(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);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,23)(13,20)(14,21)(15,17)(18,22)(19,24);(2,7)(6,8)(9,12,14,11,15,21,23,17)(10,16,18,22,20,13,19,24);(1,4,5,3)(2,8,7,6)(9,13)(10,17)(11,20)(12,18)(14,24)(15,16)(19,21)(22,23)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 18914);