Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#33443 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | C22×D4 |
| commutator | C4 | C23×C8×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C2×C53 | C2×C8×D4 |
| prime radical | C2×#135×C53 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C2×M32×C53 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,12,5,13,22,30,7,10,24,15,23,11,28,29)(3,14,17,31,18,26,32,4,9,21,27,20,16,8,25,19)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3)(2,8)(4,15)(5,18)(6,17)(7,9)(10,26)(11,21)(12,19)(13,14)(16,23)(20,29)(22,32)(24,27)(25,28)(30,31);(1,4,6,21,5,20,22,8,7,19,24,14,23,31,28,26)(2,9,12,27,13,16,30,25,10,3,15,17,11,18,29,32);(2,10)(4,19)(8,26)(11,13)(12,15)(14,21)(20,31)(29,30)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26431);