Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#25189 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| 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:
(5,6,11,15,12,16,10,26)(7,17,19,13,8,20,22,29)(9,23,24,14,25,30,27,31)(18,28,34,35,21,36,33,32)(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)(3,4)(5,7,11,19,12,8,10,22)(6,13,15,20,16,29,26,17)(9,18,24,34,25,21,27,33)(14,28,30,35,31,36,23,32);(2,4)(5,8)(6,13)(7,12)(9,21)(10,22)(11,19)(14,28)(15,17)(16,29)(18,25)(20,26)(23,35)(24,34)(27,33)(30,32)(31,36);(5,9,12,25)(6,14,16,31)(7,18,8,21)(10,27,11,24)(13,28,29,36)(15,30,26,23)(17,32,20,35)(19,34,22,33)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 18177);