Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#10279 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C73 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23×C73 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C2×C42×C73 ◁ G | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 18688 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2)(3,4)(5,6,11,8)(7,15,19,16)(9,14,10,13)(12,18,20,17)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30)(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);(2,4)(5,7,11,19)(6,12,8,20)(9,18,10,17)(13,16,14,15);(5,8,11,6)(7,16,19,15)(9,13,10,14)(12,17,20,18)(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 3267);