Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13154 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #442 |
| 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×#97×C53 ◁ G | |
314 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)(3,4)(5,6,10,15,11,17,24,16)(7,18,21,31,23,32,22,13)(8,12,25,29,20,30,26,35)(9,27,28,14,19,33,36,34)(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);(5,7,19,20,10,22,36,25,11,23,9,8,24,21,28,26)(6,12,14,32,16,29,27,13,17,30,34,18,15,35,33,31);(2,4)(5,8,9,21,10,26,28,7,11,20,19,22,24,25,36,23)(6,13,14,12,16,18,27,29,17,31,34,30,15,32,33,35)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6142);