Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#33369 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C2×#99 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C2×Q8 | C24×C53 |
| socle | C2×C53 | C22×#99 |
| prime radical | C53×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C22×#41×C53 ◁ G | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,10,7,11,21,28)(3,12,15,29,13,24,20,31)(4,9,16,26,5,17,19,27)(8,22,25,18,23,30,32,14)(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)(2,8)(4,14)(5,18)(6,20)(7,13)(9,24)(10,25)(11,23)(12,17)(15,21)(16,30)(19,22)(26,31)(27,29)(28,32);(3,13)(8,23)(12,24)(14,18)(15,20)(22,30)(25,32)(29,31);(1,4,7,5)(2,9,11,17)(3,14,13,18)(6,16,21,19)(8,24,23,12)(10,26,28,27)(15,22,20,30)(25,31,32,29)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26357);