Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#12806 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | #410 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C2×#41 | C23×C73 |
| socle | C2×C73 | #1799 |
| prime radical | #525×C73 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C2×C8×C73 ◁ C2×#131×C73 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2)(3,11)(4,15)(5,18,19,8)(6,10)(7,13)(9,17)(12,26)(14,24,28,25)(16,23,31,30)(20,32,21,29)(22,27)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3,4,13,6,12,17,27)(2,7,9,11,10,22,15,26)(5,14,16,29,19,28,31,32)(8,21,23,25,18,20,30,24);(2,8)(3,12)(7,20)(9,23)(10,18)(11,24)(13,27)(14,28)(15,30)(21,22)(25,26)(29,32)2482 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5794);