Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C28 |
| direct product | C53×#61384 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| prime radical | C2×#41×C53 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C53×#533 ◁ G | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 77 points:
(1,2,5,4)(3,7,8,6)(10,15)(11,16)(13,17)(18,23)(19,24)(20,22)(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);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,20,22)(12,13,21,17)(18,19,24,23);(9,11,14,20)(10,16,15,22)(12,18,21,24)(13,19,17,23);(1,4,5,2)(3,6,8,7)(9,12,14,21)(10,17,15,13)(11,18,20,24)(16,23,22,19);(9,13)(10,12)(11,19)(14,17)(15,21)(16,18)(20,23)(22,24)2915 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54372);