Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#8762 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #381 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C23×C4 | C23×C73 |
| prime radical | C22×D4×C73 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 97 points:
(2,5)(3,6)(7,8)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,22,24)(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);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,24)(22,23);(1,3,4,7)(2,6,5,8)(9,11)(10,15,16,22)(12,19)(13,20)(14,17,18,23)(21,24)3139 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1750);