Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C24×C4 |
| direct product | C73×#25613 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C73 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C2×#121×C73 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C2×#117×C73 ◁ G | |
1116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 101 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,22)(16,20)(17,25)(18,21)(19,23)(24,27)(26,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,2)(3,4)(6,9)(10,11)(13,14)(15,19)(16,25)(17,20)(18,21)(22,23)(24,28)(26,27);(2,4)(13,15,17,23,18,24,16,26)(14,19,20,22,21,28,25,27);(5,7,8,12)(6,10,9,11)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 18601);