Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#22249 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C22×D4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C43×C53 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,4,3)(5,6)(7,13)(8,16)(9,11)(10,14)(12,17)(15,18)(19,20)(21,22)(23,29)(24,26)(25,33)(27,32)(28,31)(30,35)(34,36)(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);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,18)(16,19)(17,20)(22,26)(23,30)(28,33)(32,36);(6,11)(7,14)(12,16)(15,20)(21,23,31,36,24,30,25,32)(22,27,33,35,26,34,28,29);(5,8,9,17)(6,12,11,16)(7,15,14,20)(10,18,13,19)(22,26)(27,34)(28,33)(29,35)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15237);