Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#18364 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,7)(3,9,10,16)(4,12,14,8)(6,13,15,11)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(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);(1,3,5,10)(2,6,7,15)(4,13,14,11)(8,16,12,9)(17,19,22,29)(18,23,25,31)(20,28,21,27)(24,32,30,26);(4,14)(8,12)(9,16)(11,13);(2,7)(3,10)(8,12)(11,13)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11352);