Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C24×C4 |
| direct product | C73×#20749 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C22×D4 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C22×C73 | C22×C22⋊C4 |
| prime radical | C22×#41×C73 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×#117×C73 ◁ G | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 97 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,12,21,16)(11,15,20,23,19,18,24,22)(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);(2,5)(6,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(9,12)(10,14)(11,18)(13,16)(15,19)(17,21)(20,22)(23,24);(1,3,4,7)(2,6,5,8)(11,19)(15,18)(20,24)(22,23)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 13737);