Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C27 |
| direct product | C73×#23401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C73×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
1954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,21)(10,14)(11,17)(13,24)(16,32)(18,30)(19,28)(20,31)(22,26)(23,29)(25,35)(27,34)(33,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(5,7,10,18)(6,11,14,26)(8,16,19,33)(9,22,24,17)(12,27,28,25)(13,30,21,15)(20,35,23,34)(29,36,31,32);(2,4)(5,8)(6,12)(7,16)(9,23)(10,19)(11,25)(13,31)(14,28)(15,32)(17,35)(18,33)(20,24)(21,29)(22,34)(26,27)(30,36);(8,19)(12,28)(16,33)(20,23)(25,27)(29,31)(32,36)(34,35)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 16389);