Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#25186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C2×2−1+4×C73 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(1,2,4,3)(5,6,10,15)(7,16,20,12)(8,21,23,30)(9,11,25,14)(13,24,29,22)(17,26,19,28)(18,33,34,36)(27,32,31,35)(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);(1,3,4,2)(6,11)(7,17)(9,25)(13,21)(14,15)(18,32)(19,20)(22,24)(26,28)(29,30)(33,36)(34,35);(2,3)(5,7,10,20)(6,12,15,16)(8,18,23,34)(9,19,25,17)(11,26,14,28)(13,27,29,31)(21,36,30,33)(22,32,24,35);(5,8)(6,13)(7,18)(9,24)(10,23)(11,21)(12,27)(14,30)(15,29)(16,31)(17,32)(19,35)(20,34)(22,25)(26,36)(28,33)3139 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 18174);