Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#17328 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C53 | C23 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| socle | C22×C53 | C22×C42 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C53 ◁ G | |
750 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32)(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);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16);(2,7)(6,12)(8,14)(13,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(19,25)(22,28)(26,31)(29,32)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10316);