Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13271 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #476 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1987 |
| prime radical | C23×C53 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,10,7,11,4,18,8,12,19,29,6,22,20,30)(3,13,16,25,15,24,32,9,17,26,14,27,23,31,21,28)(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)(2,9,12,28)(4,14)(5,21)(6,15)(7,23)(8,17)(10,24,29,31)(11,25,22,27)(13,30,26,18)(16,20)(19,32);(1,4,6,5,8,20,7,19)(2,10,11,18,12,29,22,30)(3,14,15,21,17,16,23,32)(9,24,25,13,28,31,27,26)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6259);