Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#17606 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C73×#3489 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×#41×C73 ◁ G | |
1270 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)(3,7,8,5)(4,6)(10,15,17,21)(16,22,24,23)(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)(2,5)(4,8)(6,7)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,24)(20,22);(9,11,14,19)(10,16,17,24)(12,20,13,18)(15,22,21,23);(9,12,14,13)(10,15,17,21)(11,18,19,20)(16,23,24,22)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 10594);