Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11369 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #127 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | #399 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
602 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)(7,15,19,29)(8,20)(9,13)(10,14)(11,18,26,23)(12,22)(16,24)(17,30,33,36)(21,34)(25,32,27,35)(28,31)(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);(5,7,16,18,10,19,9,23)(6,11,24,15,14,26,13,29)(8,17,31,32,22,33,21,35)(12,27,34,36,20,25,28,30);(5,8,16,31,10,22,9,21)(6,12,24,34,14,20,13,28)(7,17,18,32,19,33,23,35)(11,25,15,30,26,27,29,36)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4357);