Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9345 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#9356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #386 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,10,6,9,4,8)(3,7,11,14,12,15,13,16)(17,18,21,25,23,26,22,31)(19,27,28,32,20,29,30,24)(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);(1,3,4,11,6,12,5,13)(2,7,10,16,9,15,8,14)(17,19,23,20)(18,24,26,32)(21,30,22,28)(25,29,31,27);(1,4,6,5)(2,8,9,10)(3,11,12,13)(7,14,15,16)(17,20,23,19)(18,24,26,32)(21,30,22,28)(25,27,31,29)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2344);