Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#10786 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #386 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
1178 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)(5,6,10,16,11,14,9,15)(7,12,19,26,18,27,24,30)(8,20,22,13,23,29,21,28)(17,31,33,35,32,36,34,25)(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)(6,12,14,27)(8,17)(9,19)(10,24)(11,18)(13,25,28,35)(15,26,16,30)(20,31,29,36)(21,33)(22,34)(23,32);(5,8,11,23)(6,13,14,28)(7,17,18,32)(9,22,10,21)(12,25,27,35)(15,29,16,20)(19,34,24,33)(26,36,30,31)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3774);