Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#19085 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,7)(3,10)(4,5)(8,16)(9,14)(11,15)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26)(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);(1,2)(3,8)(4,14)(5,7)(6,9)(10,11)(12,16)(13,15)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,2)(3,11)(4,7)(5,14)(6,9)(8,10)(12,16)(13,15)(18,23)(20,30)(22,31)(26,29);(1,3)(2,8)(4,12)(5,13)(6,10)(7,15)(9,11)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12073);