Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#18942 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(18,19)(20,28)(21,22)(24,29)(25,26)(27,32)(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);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,18,23,26)(19,21,25,22)(20,24,28,32)(27,30,29,31);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,19)(18,21)(20,29)(22,26)(23,25)(24,31)(27,28)(30,32);(3,10)(7,14)(11,16)(13,15)(17,20)(18,24)(19,27)(21,30)(22,31)(23,28)(25,29)(26,32)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11930);