Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#25183 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C2×H(5,𝔽2)×C73 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ G | |
1000 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)(5,6,11,17)(7,18,22,33)(8,24,9,14)(10,28,12,16)(13,23,30,21)(15,26,32,27)(19,31,35,34)(20,29,25,36)(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);(1,3,4,2)(5,7,11,22)(6,13,17,30)(8,25,9,20)(10,23,12,21)(14,31,24,34)(15,29,32,36)(16,33,28,18)(19,27,35,26);(2,3)(5,8,11,9)(6,14,17,24)(7,19,22,35)(10,26,12,27)(13,29,30,36)(15,16,32,28)(18,34,33,31)(20,21,25,23);(5,9,11,8)(6,15,17,32)(7,20,22,25)(10,26,12,27)(13,29,30,36)(14,16,24,28)(18,31,33,34)(19,21,35,23)3139 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 18171);