Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#22691 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | D42 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ G | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,3,4)(5,6)(7,19)(8,24)(9,15)(10,17)(11,16)(12,18)(13,22)(14,25)(20,35)(21,33)(23,31)(26,32)(27,36)(28,30)(29,34)(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)(6,13)(8,20)(9,26)(10,28)(11,22)(12,23)(14,15)(16,31)(17,19)(18,30)(21,27)(24,34)(25,29)(32,36)(33,35);(5,8,10,21,12,26,11,25)(6,14,16,32,18,33,17,24)(7,20,28,27,23,9,22,29)(13,15,31,36,30,35,19,34);(5,9)(6,15)(7,21)(8,22)(10,29)(11,27)(12,20)(13,24)(14,31)(16,36)(17,34)(18,35)(19,33)(23,25)(26,28)(30,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15679);