Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#13245 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #466 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| prime radical | C22×D16×C73 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ #405×C73 ◁ G | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 109 points:
(2,4)(5,6,11,17,13,18,12,21)(7,19,22,29,15,9,16,10)(8,23,26,20,28,35,27,31)(14,30,32,36,33,24,34,25)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(1,2,3,4)(5,7,19,21,12,22,10,6,13,15,9,17,11,16,29,18)(8,20,30,34,27,35,25,14,28,31,24,32,26,23,36,33);(5,8,9,24,12,27,29,36,13,28,19,30,11,26,10,25)(6,14,7,20,17,32,22,35,18,33,15,31,21,34,16,23)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6233);