Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#9051 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #386 |
| prime radical | #121×C53 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18,21,26,22,27,20,25)(19,28,30,23,31,32,29,24)(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);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16)(17,19)(18,23)(20,30)(21,29)(22,31)(24,27)(25,32)(26,28);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)(17,19)(18,24)(20,29)(21,30)(22,31)(23,27)(25,32)(26,28)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2039);