Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#8500 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #379 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#3546 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,8,6,10,5,9)(3,11,13,15,14,16,12,7)(17,18)(19,26,28,22)(20,25)(21,24)(23,27,31,29)(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);(1,3)(2,7)(4,13)(5,12)(6,14)(8,11)(9,16)(10,15)(17,19)(18,22)(20,27)(21,28)(23,25)(24,26)(29,30)(31,32);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(17,19)(18,23)(20,29)(21,28)(22,25)(24,31)(26,32)(27,30)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1488);