Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #476 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C23×C53 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,4,3)(5,6)(7,17,20,28)(8,14)(9,25)(10,27)(11,16)(12,18,30,36)(13,23)(15,22)(19,32,21,34)(24,33)(26,31,35,29)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,3,4,2)(5,7,10,19,11,20,22,21)(6,12,15,29,16,30,27,31)(8,18,23,35,24,36,9,26)(13,28,14,32,25,17,33,34);(5,8,22,9,11,24,10,23)(6,13,27,33,16,25,15,14)(7,18,21,26,20,36,19,35)(12,28,31,34,30,17,29,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6175);