Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C73 · Schur multiplier (H2) = C27×C4 |
| direct product | C73×#60128 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C25 |
| commutator | C22 | C24×C4×C73 |
| Frattini, cosocle | C2×C4 | C25×C73 |
| socle | C22×C73 | C24×C4 |
| prime radical | C22×#41×C73 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ G | |
2124 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)(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24)(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);(1,3,4,7)(2,6,5,8)(10,14)(15,20)(16,21)(22,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,20)(18,23)(22,24)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53116);