Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C24 |
| direct product | C73×#13005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | #1874 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| prime radical | C73×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C22×C4×C73 ◁ C2×#131×C73 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,6,11,7,13,22,25,8,14,4,20,5,12,21,9)(3,15,18,24,17,28,32,29,19,30,16,31,23,10,26,27)(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);(2,9,14,25)(4,21)(5,7)(6,22)(10,24,28,31)(11,13,20,12)(15,27,30,29)(16,26)(17,23)(18,32);(1,3)(2,10)(4,18)(5,23)(6,16)(7,17)(8,19)(9,24)(11,29)(12,30)(13,15)(14,28)(20,27)(21,32)(22,26)(25,31)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5993);