Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#33659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C53 |
| prime radical | C53×#3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,7)(4,14)(5,17)(8,20)(9,19)(10,16)(12,26)(13,25)(15,29)(18,21)(22,30)(23,27)(24,32)(28,31)(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);(3,11)(4,15)(7,19)(8,22)(13,25)(14,29)(16,21)(23,28);(1,2,5,18,6,9,17,10)(3,8,13,28,11,22,25,23)(4,16,14,19,15,21,29,7)(12,27,26,30,24,31,32,20);(1,3,6,11)(2,8,9,22)(4,12,15,24)(5,13,17,25)(7,20,19,30)(10,23,18,28)(14,26,29,32)(16,27,21,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26647);