Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#22483 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×C53×#555 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C4×D4×C53 ◁ G | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 77 points:
(2,5)(6,8)(9,10,13,18,14,17,11,15)(12,20,19,24,21,16,22,23)(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);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24);(2,5)(6,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)2915 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15471);