Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C27×C4 |
| direct product | C73×#60620 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| prime radical | C22×#41×C73 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×#117×C73 ◁ G | |
2014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 5 generators on 113 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,21,23,32)(12,26,27,34)(13,19,14,18)(16,25,31,24)(17,28,33,29)(22,30,36,40)(35,37,39,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,5)(6,8)(9,11,14,25,15,23,13,24)(10,16,19,32,20,31,18,21)(12,22,29,38,27,36,28,37)(17,30,26,39,33,40,34,35);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,27,14,29)(10,17,18,34,20,33,19,26)(11,22,24,37,23,36,25,38)(16,30,21,35,31,40,32,39);(12,27)(17,33)(22,36)(26,34)(28,29)(30,40)(35,39)(37,38);(11,23)(16,31)(21,32)(22,36)(24,25)(30,40)(35,39)(37,38)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53608);