Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C29 |
| direct product | C53×#61100 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| socle | C2×C53 | C2×D42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
2616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 5 generators on 85 points:
(2,8)(5,6)(9,21)(10,22)(11,19)(12,24)(14,20)(16,17)(23,31)(25,29)(26,32)(27,30)(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);(1,2)(3,12)(4,10)(5,19)(6,8)(7,11)(9,14)(13,26)(15,21)(16,29)(17,22)(18,25)(20,24)(23,27)(28,31)(30,32);(1,2)(3,9)(4,10)(5,8)(6,19)(7,11)(12,14)(13,23)(15,24)(16,22)(17,29)(18,25)(20,21)(26,27)(28,32)(30,31);(1,3)(2,9)(4,13)(5,20)(6,14)(7,15)(8,21)(10,23)(11,24)(12,19)(16,30)(17,27)(18,28)(22,31)(25,32)(26,29);(1,4,7,18)(2,10,11,25)(3,13,15,28)(5,16,6,17)(8,22,19,29)(9,23,24,32)(12,26,21,31)(14,27,20,30)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54088);