Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#29155 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | #2563 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,13,16,9)(4,17)(5,8)(6,11)(7,12)(10,19)(14,25,21,22)(15,29,31,32)(18,23)(20,26)(24,28,27,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(2,8,12,11)(3,14,16,21)(4,18)(9,13)(10,17,26,23)(15,30,31,28)(19,20)(22,25);(1,3)(2,9)(4,15)(5,14)(6,21)(7,16)(8,22)(10,24)(11,25)(12,13)(17,32)(18,31)(19,28)(20,30)(23,29)(26,27);(1,4,5,19,7,18,6,20)(2,10,11,17,12,26,8,23)(3,15,14,28,16,31,21,30)(9,24,25,32,13,27,22,29)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 22143);