Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C27 |
| direct product | C73×#23403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C2×#575×C73 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
1942 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)(7,16)(8,13)(9,14)(10,23)(11,15)(12,18)(17,32)(19,26)(20,29)(21,31)(22,30)(24,27)(25,35)(28,33)(34,36)(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)(6,12)(8,17)(9,18)(10,24)(11,19)(13,28)(14,26)(15,27)(16,23)(20,33)(21,35)(22,34)(25,30)(29,36)(31,32);(2,4)(5,8,11,22)(6,13,15,30)(7,17,19,34)(9,21,10,20)(12,25,27,28)(14,31,23,29)(16,32,26,36)(18,35,24,33);(5,9,11,10)(6,14,15,23)(7,18,19,24)(8,20,22,21)(12,26,27,16)(13,29,30,31)(17,33,34,35)(25,32,28,36)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 16391);