Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#19093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | #121×C53 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 117 points:
(1,2,6,9)(3,11,13,25)(4,8,16,23)(5,17,20,10)(7,18,22,14)(12,28,29,27)(15,26,19,24)(21,31,32,30)(33,34)(35,39)(36,46)(37,49)(38,41)(40,48)(42,52)(43,59)(44,57)(45,54)(47,58)(50,55)(51,56)(53,61)(60,63)(62,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,3,6,13)(2,7,9,22)(4,12,16,29)(5,18,20,14)(8,21,23,32)(10,25,17,11)(15,31,19,30)(24,27,26,28)(33,35,38,45)(34,39,41,54)(36,43,47,60)(37,50,52,44)(40,53,56,64)(42,57,49,55)(46,63,58,59)(48,62,51,61);(1,4,6,16)(2,8,9,23)(3,12,13,29)(5,19,20,15)(7,21,22,32)(10,26,17,24)(11,27,25,28)(14,31,18,30)(33,36)(34,40)(35,43)(37,51)(38,47)(39,53)(41,56)(42,58)(44,62)(45,60)(46,49)(48,52)(50,61)(54,64)(55,63)(57,59);(2,9)(3,13)(8,23)(10,17)(12,29)(14,18)(24,26)(30,31)(34,41)(39,54)(40,56)(42,49)(46,58)(53,64)(55,57)(59,63)1643 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12081);