Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#24985 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | #2420 |
| prime radical | C53×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
646 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)(3,15)(4,19)(5,9)(6,13)(7,12)(8,14)(10,21)(11,18)(16,30)(17,25)(20,29)(22,27)(23,28)(24,31)(26,32)(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);(1,3,7,17,8,18,6,5)(2,9,13,11,14,25,12,15)(4,20,21,31,22,32,23,16)(10,26,19,24,28,29,27,30);(1,4)(2,10)(3,16)(5,20)(6,21)(7,23)(8,22)(9,24)(11,26)(12,19)(13,27)(14,28)(15,29)(17,32)(18,31)(25,30);(1,5,6,18,8,17,7,3)(2,11,12,9,14,15,13,25)(4,20,21,31,22,32,23,16)(10,26,19,24,28,29,27,30)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 17973);