Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11356 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #390 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | #1445 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
580 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)(4,8,17,24)(5,18)(6,10)(7,14)(9,12)(13,28,30,26)(15,25)(16,31,27,23)(19,21)(20,29,22,32)(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);(1,3,12,14,6,15,5,19)(2,7,9,25,10,21,18,11)(4,13,27,29,17,30,16,32)(8,22,23,26,24,20,31,28);(1,4,12,27,6,17,5,16)(2,8,9,23,10,24,18,31)(3,13,14,29,15,30,19,32)(7,20,25,28,21,22,11,26)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4344);