Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#29454 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C53×#376 ◁ G | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,7)(4,6)(5,8)(17,18,21,25,22,26,30,32)(19,24,27,31,20,28,29,23)(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);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(19,20)(23,31)(24,28)(27,29);(10,13)(14,16)(17,19,22,20)(18,23,26,31)(21,29,30,27)(24,25,28,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,24,26,28)(21,29,30,27)(23,25,31,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 22442);