Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#9782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #390 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C23×C4 | C23×C73 |
| prime radical | C24×C73 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2)(3,5,7,8)(4,6)(9,10)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(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)(2,5,6,8)(4,7)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24);(2,6)(5,8)(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 2770);