Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#19567 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C23×C53 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(3,9)(4,11)(6,13)(8,15)(10,12)(14,16)(17,18,22,24)(19,27,30,31)(20,25,23,32)(21,29,28,26)(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)(3,6)(4,8)(5,7)(9,13)(10,14)(11,15)(12,16)(17,19,28,32,22,30,21,25)(18,23,29,31,24,20,26,27);(1,3,5,10)(2,6,7,14)(4,12,11,9)(8,16,15,13)(18,24)(20,23)(26,29)(27,31);(2,7)(6,14)(8,15)(13,16)(17,20)(18,25)(19,29)(21,31)(22,23)(24,32)(26,30)(27,28)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12555);