Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#358×C73 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C73 | D8 |
| commutator | C8 | C2×C42×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle, prime radical | C23×C73 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 18688 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2,3,4)(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);(5,6,7,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5624);