Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#18019 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C22×D4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| socle | C22×C53 | C2×C4×D4 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C53×#431 ◁ G | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28)(20,25)(21,26)(22,31)(23,27)(24,29)(30,32)(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);(17,19,23,21)(18,24,27,32)(20,29,22,30)(25,26,31,28);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(17,20,23,22)(18,25,27,31)(19,29,21,30)(24,26,32,28);(3,9)(6,12)(10,15)(13,16)(17,21,23,19)(18,26,27,28)(20,29,22,30)(24,31,32,25)3710 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11007);