Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13323 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(2,8)(3,12)(4,17)(5,6)(9,25)(10,29)(11,24)(13,23)(14,19)(15,16)(18,22)(20,21)(26,27)(28,31)(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);(1,2)(3,13)(4,18)(5,10)(6,22)(7,11)(8,17)(9,15)(12,28)(14,26)(16,27)(19,23)(20,24)(21,29)(25,32)(30,31);(1,3)(2,9)(4,19)(5,15)(6,12)(7,16)(8,23)(10,27)(11,28)(13,22)(14,21)(17,32)(18,31)(20,30)(24,26)(25,29)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6311);