Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#7600 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C73 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| prime radical | C24×C73 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C22×C42×C73 ◁ G | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,15,13,16,11,14)(17,18,21,23,22,19,20,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,3,4,8)(2,5,6,7)(17,19)(18,22)(20,24)(21,23);(9,11,13,12)(10,14,16,15)(17,20,22,21)(18,23,19,24)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 588);