Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#33457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×D8 |
| commutator | C8 | C23×C4×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| prime radical | C22×D8×C53 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ #117×C53 ◁ C53×#405 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,4,3)(6,11)(7,15)(8,18)(9,20)(12,26)(13,29)(14,25)(16,32)(19,22)(21,23)(24,27)(28,33)(30,34)(35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,6,10,14)(7,12,15,27)(8,19,21,29)(9,22,20,13)(11,23,25,18)(16,33,30,36)(17,35,31,28)(24,32,26,34);(1,3,4,2)(7,15)(12,27)(16,30)(17,31)(24,26)(28,35)(32,34)(33,36);(5,7)(6,12)(8,16)(9,17)(10,15)(11,24)(13,28)(14,27)(18,34)(19,33)(20,31)(21,30)(22,35)(23,32)(25,26)(29,36)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26445);