Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11334 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #392 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | #1444 |
| prime radical | C24×C53 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,10,7,11,5,21)(3,12,16,23,15,27,14,8)(4,9,20,24,19,25,18,31)(13,28,30,22,17,26,29,32)(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);(1,3,5,14,7,15,6,16)(2,8,10,12,11,23,21,27)(4,17,18,30,19,13,20,29)(9,22,24,26,25,32,31,28);(1,4)(2,9)(3,13,15,17)(5,18)(6,20)(7,19)(8,22,23,32)(10,24)(11,25)(12,26,27,28)(14,29,16,30)(21,31)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4322);