Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#19004 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,7)(4,5)(8,15)(9,12)(10,13)(14,16)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32)(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);(1,2,6,9)(3,8,11,16)(4,7,5,12)(10,14,13,15)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,27)(28,30);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11992);