Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#33463 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | C2×D8 |
| commutator | C8 | C23×C4×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C2×C53 | C2×C4×D8 |
| prime radical | C53×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,12,7,10,23,13)(3,14,17,30,19,32,18,26)(4,9,5,21,20,11,24,29)(8,25,28,15,27,22,31,16)(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)(5,22)(6,17)(7,19)(9,26)(10,27)(11,30)(12,28)(13,31)(14,21)(16,20)(18,23)(24,25)(29,32);(1,4,6,5,7,20,23,24)(2,9,12,21,10,11,13,29)(3,15,17,22,19,16,18,25)(8,26,28,14,27,30,31,32);(2,10)(8,27)(9,11)(12,13)(14,32)(21,29)(26,30)(28,31)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26451);