Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#22690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | D42 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ G | |
846 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)(7,19)(8,24)(9,15)(10,17)(11,16)(12,18)(13,22)(14,21)(20,34)(23,35)(25,31)(26,32)(27,30)(28,36)(29,33)(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);(5,7,12,23)(6,13,18,31)(8,20,27,9)(10,25,11,22)(14,15,32,34)(16,35,17,19)(21,29,26,28)(24,33,30,36);(5,8,11,21,12,27,10,26)(6,14,17,24,18,32,16,30)(7,20,22,29,23,9,25,28)(13,15,19,33,31,34,35,36);(5,9)(6,15)(7,21)(8,25)(10,29)(11,28)(12,20)(13,30)(14,19)(16,36)(17,33)(18,34)(22,27)(23,26)(24,31)(32,35)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15678);