Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#25183 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×H(5,𝔽2)×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ G | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,3,4)(5,6,11,16)(7,18,22,32)(8,23,9,13)(10,17,28,15)(12,27,31,21)(14,25,34,26)(19,35,36,33)(20,30,24,29)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,3,4)(5,7,11,22)(6,12,16,31)(8,24,9,20)(10,27,28,21)(13,33,23,35)(14,30,34,29)(15,32,17,18)(19,25,36,26);(1,2)(3,4)(5,8,11,9)(6,13,16,23)(7,19,22,36)(10,25,28,26)(12,29,31,30)(14,15,34,17)(18,33,32,35)(20,27,24,21);(5,9,11,8)(6,14,16,34)(7,20,22,24)(10,25,28,26)(12,30,31,29)(13,15,23,17)(18,33,32,35)(19,27,36,21)2279 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 18171);