Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#11031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #392 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
1262 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)(7,16)(8,21,24,13)(9,25)(10,14)(11,15)(12,18)(17,29,34,33)(19,28)(20,32)(22,36,35,27)(23,30,26,31)(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);(5,7,11,20)(6,12,15,28)(8,22,24,35)(9,19,10,18)(13,29,21,33)(14,32,25,16)(17,26,34,23)(27,30,36,31);(5,8)(6,13)(7,17)(9,23)(10,26)(11,24)(12,27)(14,30)(15,21)(16,33)(18,22)(19,35)(20,34)(25,31)(28,36)(29,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4019);