Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C24 |
| direct product | C73×#13068 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | #1893 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| prime radical | C2×#472×C73 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C22×C4×C73 ◁ C2×#131×C73 ◁ G | |
470 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)(3,12)(4,17,20,28)(5,15)(6,10)(7,14)(8,11)(9,19,27,25)(13,18,26,21)(16,22)(23,24)(29,30,32,31)(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);(1,3,12,2,7,15,23,10,8,16,22,11,6,24,5,14)(4,13,29,9,19,31,18,28,20,26,32,27,25,30,21,17);(1,4)(2,9)(3,13)(5,21)(6,25)(7,19)(8,20)(10,28)(11,27)(12,29)(14,17)(15,31)(16,26)(18,23)(22,32)(24,30)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6056);