Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#8486 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #379 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#3559 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,9,6,10,4,14)(3,11,12,7,13,8,15,16)(18,22,24,27)(23,31,32,30)(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)(4,15)(5,12)(6,13)(8,9)(10,16)(11,14)(17,18)(19,23)(20,27)(21,24)(22,29)(25,30)(26,32)(28,31);(1,3)(2,8)(4,12)(5,15)(6,13)(7,9)(10,11)(14,16)(17,19,21,26)(18,23,24,32)(20,28,29,25)(22,30,27,31)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1474);