Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11336 |
| 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 | |
694 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)(5,6,10,14,11,16,9,15)(7,17,21,27,20,28,19,12)(8,13,23,31,25,32,24,30)(18,29,34,26,35,33,22,36)(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);(5,7,9,19,11,20,10,21)(6,12,14,17,16,27,15,28)(8,22,23,18,25,34,24,35)(13,29,30,36,32,33,31,26);(5,8)(6,13)(7,18,20,35)(9,24)(10,23)(11,25)(12,26,27,36)(14,31)(15,30)(16,32)(17,33,28,29)(19,22,21,34)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4324);