Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#22694 |
| 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 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,12,7,13,24,11)(3,14,18,27,19,26,17,8)(4,20,16,9,21,28,32,29)(5,10,23,30,15,25,22,31)(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)(2,8)(4,15)(5,21)(6,18)(7,19)(9,25)(10,29)(11,26)(12,14)(13,27)(16,22)(17,24)(20,30)(23,32)(28,31);(1,4,6,16,7,21,24,32)(2,9,11,20,13,29,12,28)(3,15,18,22,19,5,17,23)(8,25,26,30,27,10,14,31);(1,5)(2,10)(3,16)(4,17)(6,23)(7,15)(8,20)(9,14)(11,31)(12,30)(13,25)(18,21)(19,32)(22,24)(26,29)(27,28)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15682);