Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#32699 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×AGL(1,ℤ/8)×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,7,11,6,10,22,12)(3,4,8,16,14,17,23,27)(5,18,21,28,20,9,24,26)(13,15,29,30,19,31,25,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3)(2,4)(5,19)(6,14)(7,23)(8,22)(9,15)(10,17)(11,27)(12,16)(13,20)(18,31)(21,29)(24,25)(26,32)(28,30);(1,4,6,17)(2,8)(3,12)(5,15,20,31)(7,16,22,27)(9,25)(10,23)(11,14)(13,26)(18,29)(19,28)(21,30,24,32);(1,5,6,20)(2,9,10,18)(3,13,14,19)(4,15,17,31)(7,24,22,21)(8,25,23,29)(11,28,12,26)(16,32,27,30)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 25687);