Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#10035 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #392 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,10)(3,11,14,15)(4,9,5,8)(7,13,16,12)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(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);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(22,31)(25,32);(3,12)(4,5)(7,11)(8,9)(13,14)(15,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3023);