Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C24 |
| direct product | C73×#23753 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C2×C73 | C2×#400 |
| prime radical | C24×C73 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C23×C4×C73 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 4 generators on 105 points:
(1,2,5,10,6,11,7,22)(3,12,14,24,16,27,15,8)(4,17,20,31,19,9,21,26)(13,25,29,32,28,30,18,23)(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);(1,3,7,14,6,16,5,15)(2,8,22,12,11,24,10,27)(4,18,21,13,19,29,20,28)(9,25,31,32,17,30,26,23);(1,4,5,20,6,19,7,21)(2,9,10,26,11,17,22,31)(3,13,14,29,16,28,15,18)(8,23,12,25,24,32,27,30);(4,19)(9,17)(13,28)(18,29)(20,21)(23,32)(25,30)(26,31)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 16741);