Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #476 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,3,4)(5,6)(7,15)(8,21,24,35)(9,26)(10,27)(11,16)(12,19)(13,23,30,18)(14,17)(20,33)(22,36,25,32)(28,29,34,31)(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);(5,7,17,10,11,20,9,19)(6,12,26,33,16,27,14,15)(8,18,25,28,24,23,22,34)(13,21,31,36,30,35,29,32);(5,8)(6,13)(7,18)(9,22)(10,28)(11,24)(12,21)(14,29)(15,32)(16,30)(17,25)(19,34)(20,23)(26,31)(27,35)(33,36)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5812);