Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#18668 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C73 |
| prime radical | C73×#3245 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 97 points:
(2,6)(3,4)(7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(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);(1,2,5,8)(3,6,4,7)(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,23)(20,24);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 11656);