Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C53×#1991 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C2×D4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ C22×C42×C53 ◁ G | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 83 points:
(1,2)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(3,4,6,5)(7,8,10,12)(9,13,14,11)(15,16)(17,23)(18,27)(19,21)(20,25)(22,24)(26,29)(28,30);(3,5,6,4)(15,17,24,30,19,26,18,25)(16,20,27,29,21,28,22,23);(7,9,10,14)(8,11,12,13)(16,21)(20,28)(22,27)(23,29)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 14305);