Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C28 |
| direct product | C53×#60600 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C4×#41×C53 ◁ G | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 77 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,13)(11,18,21,24)(12,17,15,22)(16,23,20,19)(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)(10,13)(16,20)(17,22)(18,24);(9,11,14,21)(10,16,13,20)(12,19,15,23)(17,24,22,18);(1,3,4,7)(2,6,5,8)(9,12,14,15)(10,17,13,22)(11,19,21,23)(16,24,20,18);(9,13,14,10)(11,20,21,16)(12,17,15,22)(18,19,24,23)3233 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53588);