Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#24775 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C2×AGL(1,ℤ/8)×C73 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(2,3)(5,6,10,15,9,14,11,16)(7,17,20,32,19,12,26,28)(8,21,24,30,23,31,25,13)(18,34,22,27,35,29,36,33)(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);(1,2,4,3)(5,7,9,19)(6,12,14,17)(8,22,23,36)(10,20,11,26)(13,29,30,34)(15,28,16,32)(18,24,35,25)(21,33,31,27);(5,8,9,23)(6,13,14,30)(7,18,19,35)(10,25,11,24)(12,27,17,33)(15,31,16,21)(20,36,26,22)(28,34,32,29);(1,3,4,2)(6,14)(12,17)(13,30)(15,16)(21,31)(27,33)(28,32)(29,34)3139 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 17763);