Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C27×C4 |
| direct product | C73×#60621 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| prime radical | C73×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×#117×C73 ◁ G | |
2078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 5 generators on 97 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,14)(13,17)(15,18)(16,20)(21,23)(22,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(2,5)(6,8)(9,11,14,22,15,21,13,20)(10,16,17,23,18,24,12,19);(1,3,4,7)(2,6,5,8)(9,12,13,18,15,17,14,10)(11,19,20,24,21,23,22,16);(9,13,15,14)(10,17,18,12)(11,20,21,22)(16,23,24,19);(11,21)(16,24)(19,23)(20,22)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53609);