Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#11643 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C22×C53 | C2×C22⋊C8 |
| prime radical | C23×C53 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 101 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,21,27,23,25,37,41,22,26,36,42,20,32,34,45)(19,28,30,43,29,38,44,47,31,24,35,39,33,40,46,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19,22,31)(18,24,26,28)(20,33,23,29)(21,35,36,30)(25,40,32,38)(27,43,42,39)(34,44,37,46)(41,47,45,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,22,23)(18,25,26,32)(19,29,31,33)(21,34,36,37)(24,38,28,40)(27,41,42,45)(30,44,35,46)(39,47,43,48)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4631);