Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#22245 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle, prime radical | C23×C53 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×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:
(2,4)(5,6)(7,13)(8,16)(9,11)(10,14)(12,17)(15,18)(19,20)(21,22,27,23,26,29,25,31)(24,28,33,30,34,36,35,32)(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)(6,10)(8,15)(9,14)(11,13)(12,18)(16,19)(17,20)(21,23,27,29,26,31,25,22)(24,32,33,28,34,30,35,36);(1,2,3,4)(6,11)(7,14)(12,16)(15,20)(21,24,26,34)(22,28,29,36)(23,30,31,32)(25,35,27,33);(5,8,9,17)(6,12,11,16)(7,15,14,20)(10,18,13,19)(21,25,26,27)(22,23,29,31)(24,33,34,35)(28,32,36,30)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15233);