Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#18269 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | #525×C73 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 105 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(18,23)(20,21)(24,31)(25,29)(26,27)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,30)(28,32);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19,22,28)(18,24,25,32)(20,27,21,26)(23,30,29,31);(17,20,22,21)(18,23,25,29)(19,26,28,27)(24,31,32,30)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 11257);