Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C53×#13088 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#99 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×D4 | C23×C53 |
| prime radical | C53×#2433 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C24×C53 ◁ C23×D4×C53 ◁ G | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 65 points:
(1,2)(3,4)(5,6)(7,8,10,12)(9,11)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(5,7,12,6)(8,11,9,10);(2,4)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6076);