Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#13395 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #2034 |
| prime radical | C53×#3050 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C23×C8×C53 ◁ G | |
422 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)(3,11)(4,15)(5,20)(6,9)(7,13)(8,18)(10,17)(12,27)(14,24)(16,23)(19,31)(21,28)(22,29)(25,32)(26,30)(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);(7,22)(11,27)(23,26)(24,28);(1,3,5,21,6,12,17,14)(2,7,10,26,9,22,20,23)(4,16,18,29,19,30,32,13)(8,24,15,27,25,28,31,11)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6383);