Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#13197 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C73 | #121 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C2×C73 | #1954 |
| prime radical | C2×#135×C73 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C73 ◁ C22×C8×C73 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,6,13,7,12,23,28,8,14,22,10,5,11,4,21)(3,15,19,31,17,27,16,30,20,29,24,32,18,9,25,26)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3,4,16,5,18,22,19,8,20,23,25,7,17,6,24)(2,9,21,31,11,29,10,26,14,27,28,32,12,15,13,30);(1,4,5,22,8,23,7,6)(2,10,12,21,14,13,11,28)(3,16,18,19,20,25,17,24)(9,26,15,31,27,30,29,32)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6185);