Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×C53×#2110 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C42 |
| commutator | C4 | C22×C42×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C22×C53 | C2×#97 |
| prime radical | C24×C53 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ G | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 87 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(3,4,6,10,7,12,18,26,8,11,19,25,20,27,33,34)(5,13,15,29,16,22,31,24,14,28,17,32,30,9,21,23);(3,5)(4,9,11,22)(6,17)(7,16)(8,14)(10,24,25,23)(12,13,27,28)(15,19)(18,21)(20,30)(26,32,34,29)(31,33);(5,14)(9,22)(13,28)(15,17)(16,30)(21,31)(23,24)(29,32)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26316);