Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#13294 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#99 |
| commutator | C24 | C22×C4×C73 |
| Frattini, cosocle | C22×D4 | C23×C73 |
| prime radical | C22×D4×C73 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ C2×C4×D4×C73 ◁ G | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,17,19,24)(12,20,22,16)(13,15)(14,18)(21,23)(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)(6,7)(9,11,12,21,13,19,22,18)(10,14,16,17,15,23,20,24);(1,3,4,8)(2,6,5,7)(10,15)(14,23)(16,20)(17,24)2701 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6282);