Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#19573 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,16)(17,18,22,24)(19,27,30,31)(20,25,23,32)(21,29,28,26)(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);(1,2,7,14,5,9,4,6)(3,8,10,16,11,15,12,13)(17,19,28,32,22,30,21,25)(18,23,29,31,24,20,26,27);(1,3,5,11)(2,8,9,15)(4,12,7,10)(6,13,14,16)(18,24)(20,23)(26,29)(27,31);(3,11)(8,15)(10,12)(13,16)(17,20)(18,25)(19,29)(21,31)(22,23)(24,32)(26,30)(27,28)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12561);