Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#21733 |
| 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×#97×C53 ◁ G | |
654 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)(9,10,15,20,16,21,14,31)(11,17,25,33,26,34,24,39)(12,27,13,19,29,36,28,18)(22,37,30,35,38,40,23,32)(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)(10,17)(12,22)(13,30)(14,24)(15,25)(16,26)(18,32)(19,35)(20,33)(21,34)(23,28)(27,37)(29,38)(31,39)(36,40);(9,12,14,28,16,29,15,13)(10,18,20,27,21,19,31,36)(11,22,24,23,26,38,25,30)(17,32,33,37,34,35,39,40);(9,13,15,29,16,28,14,12)(10,19,20,36,21,18,31,27)(11,23,25,22,26,30,24,38)(17,32,33,37,34,35,39,40)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 14721);