Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#9036 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #386 |
| prime radical | C53×#392 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,8,5,7,4,6)(9,10,13,17,14,11,15,19)(12,18,21,24,20,16,22,23)(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);(1,2,5,7)(3,6,4,8)(9,11,13,19,14,10,15,17)(12,18,22,23,20,16,21,24);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,24)(19,23)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2024);