Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#22695 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | D42 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | D4×C4⋊C4 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,19,27,21,28)(4,20,17,10,23,26,22,30)(5,11,24,32,16,29,25,31)(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)(2,9)(4,5)(6,21)(7,18)(8,19)(10,29)(11,30)(12,27)(13,15)(14,28)(16,23)(17,24)(20,32)(22,25)(26,31);(1,4,7,17,8,23,6,22)(2,10,13,20,14,30,12,26)(3,16,18,25,19,5,21,24)(9,11,15,31,28,29,27,32);(1,5)(2,11)(3,17)(4,21)(6,25)(7,24)(8,16)(9,26)(10,15)(12,32)(13,31)(14,29)(18,23)(19,22)(20,28)(27,30)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15683);