Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#21737 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C22×D4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C22×C53 | C2×C4×D4 |
| prime radical | C23×C53 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 93 points:
(1,2,4,6)(3,7,8,5)(9,10,14,19,15,20,29,18)(11,16,23,33,24,34,39,32)(12,25,13,17,27,35,28,36)(21,37,38,30,26,40,22,31)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,31)(18,32)(19,33)(20,34)(21,27)(25,40)(28,38)(29,39)(30,36)(35,37);(9,12,14,13,15,27,29,28)(10,17,18,25,20,36,19,35)(11,21,23,38,24,26,39,22)(16,30,32,37,34,31,33,40);(9,13,14,27,15,28,29,12)(10,17,19,35,20,36,18,25)(11,22,23,21,24,38,39,26)(16,31,33,37,34,30,32,40)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 14725);