Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#13087 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#99 |
| commutator | C24 | C22×C4×C53 |
| Frattini, cosocle | C22×D4 | C23×C53 |
| prime radical | C53×#2404 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×#41×C53 ◁ G | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2)(3,4)(5,6,9,7,10,14,11,13)(8,12,16,15,17,20,18,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(5,7)(6,11,14,9)(8,15)(10,13)(12,18,20,16)(17,19);(2,4)(5,8,10,17)(6,12,14,20)(7,15,13,19)(9,16,11,18)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6075);