Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C28×C4 |
| direct product | C53×#60643 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| socle | C22×C53 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×#117×C53 ◁ G | |
3194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 5 generators on 89 points:
(2,4)(5,6)(7,16)(8,13)(9,14)(10,24)(11,15)(12,19)(17,29)(18,26)(20,30)(21,27)(22,28)(23,34)(25,33)(31,36)(32,35)(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);(5,7,10,19,11,20,9,18)(6,12,14,16,15,26,24,30)(8,17,23,33,21,31,22,32)(13,25,28,29,27,35,34,36);(1,2,3,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,21)(12,25)(14,28)(15,27)(16,29)(18,32)(19,33)(20,31)(24,34)(26,35)(30,36);(8,21)(13,27)(17,31)(22,23)(25,35)(28,34)(29,36)(32,33);(5,9,11,10)(6,14,15,24)(7,18,20,19)(8,22,21,23)(12,16,26,30)(13,28,27,34)(17,32,31,33)(25,29,35,36)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53631);