Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C27 |
| direct product | C73×#34902 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C73 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C73 |
| socle | C23×C73 | C25 |
| prime radical | C22×D4×C73 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C73 ◁ G | |
1768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 5 generators on 97 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(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)(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,21)(22,24);(9,12,13,11)(10,14,15,16)(18,21)(22,23);(17,19,20,24)(18,22,21,23);(1,3,4,7)(2,6,5,8)5548 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 27890);