Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#23047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | D42 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×#41×C53 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ G | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,4,3)(5,6,10,16,11,17,9,15)(7,18,21,12,22,29,20,13)(8,23,26,32,27,14,25,33)(19,34,35,30,36,31,24,28)(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);(1,3,4,2)(5,7)(6,12)(8,24)(9,21)(10,20)(11,22)(13,17)(14,31)(15,29)(16,18)(19,25)(23,34)(26,36)(27,35)(28,32)(30,33);(5,7)(6,13)(8,19)(9,20)(10,21)(11,22)(12,17)(14,30)(15,29)(16,18)(23,28)(24,25)(26,35)(27,36)(31,33)(32,34);(5,8,9,25,11,27,10,26)(6,14,15,32,17,23,16,33)(7,19,20,24,22,36,21,35)(12,28,18,31,13,30,29,34)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16035);