Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#33401 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | C22×D4 |
| commutator | C4 | C23×C8×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C2×C53 | C22×C4⋊C8 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C2×#131×C53 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,9,7,11,3,13,8,12,15,23,5,21,16,27)(4,10,19,22,20,25,14,26,17,24,29,31,18,30,28,32)(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);(1,3,5,6,8,16,7,15)(2,9,11,13,12,23,21,27)(4,14,18,19,17,28,20,29)(10,22,25,26,24,31,30,32);(1,4)(2,10)(3,14)(5,18)(6,19)(7,20)(8,17)(9,22)(11,25)(12,24)(13,26)(15,29)(16,28)(21,30)(23,31)(27,32);(4,17)(10,24)(14,28)(18,20)(19,29)(22,31)(25,30)(26,32)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26389);