Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#32702 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | #2782 |
| prime radical | C53×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,10,7,12,5,11)(3,13,16,8,17,24,15,25)(4,18,21,9,22,27,20,28)(14,29,30,26,31,23,19,32)(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)(2,8)(4,19)(5,15)(6,16)(7,17)(9,26)(10,24)(11,13)(12,25)(14,21)(18,29)(20,31)(22,30)(23,27)(28,32);(1,4,5,20,7,22,6,21)(2,9,10,27,12,28,11,18)(3,14,16,30,17,31,15,19)(8,23,13,26,25,29,24,32);(3,15,17,16)(8,24,25,13)(14,19,31,30)(23,32,29,26)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 25690);