Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C28 |
| direct product | Q16×#41×C53 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| socle | C22×C53 | C23×D4 |
| prime radical | C2×#41×C53 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C4×#41×C53 ◁ G | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 77 points:
(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18)(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);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8);(1,3,4,7)(2,6,5,8)3710 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54417);