Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10690 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#10690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #401 |
| commutator, Frattini, cosocle | C25 | C23×C53 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C25×C53 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,9,11,6)(4,7,13,16)(5,8)(10,15)(12,14)(17,18)(19,25)(20,28,30,23)(21,24)(22,27)(26,32,29,31)(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)(2,6)(4,12,13,10)(5,11)(7,15,16,14)(8,9)(17,19)(18,22,24,25)(20,29)(21,27)(23,31,28,32)(26,30);(1,4)(2,7,8,16)(3,10)(5,13)(6,14,9,15)(11,12)(17,20)(18,23)(19,26,27,29)(21,30)(22,31,25,32)(24,28)1325 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3678);