Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12945 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #447 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,4,3)(5,6)(7,17)(8,12)(9,14)(10,26)(11,16)(13,19)(15,22)(18,34)(20,28)(21,35)(23,30)(24,32)(25,29)(27,31)(33,36)(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)(6,12)(8,9)(10,22)(13,28)(14,15)(16,30)(17,34)(18,36)(19,24)(20,27)(23,25)(26,29)(31,33)(32,35);(5,7)(6,13)(8,19)(9,24)(10,27)(11,21)(12,28)(14,17)(15,34)(16,33)(18,25)(20,22)(23,36)(26,32)(29,35)(30,31)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5933);