Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#18086 |
| 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 | C23×C53 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C53×#431 ◁ G | |
572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,9,6,3,7,11)(4,8,13,15,12,10,14,16)(17,18,21,25,22,24,20,29)(19,23,27,31,28,30,26,32)(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);(4,12)(8,10)(13,14)(15,16)(17,18,22,24)(19,23,28,30)(20,29,21,25)(26,32,27,31);(1,3,5,11,6,2,7,9)(4,8,13,15,12,10,14,16)(17,19,20,26,22,28,21,27)(18,23,29,32,24,30,25,31);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(18,24)(23,30)(25,29)(31,32)3710 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11074);