Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C26 |
| direct product | C2×C73×#2031 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C22×C73 | #393 |
| prime radical | C22×#41×C73 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C23×C4×C73 ◁ G | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 4 generators on 91 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(3,4)(5,12,14,18)(6,10)(7,16)(8,11)(9,15,17,13);(5,13,14,15)(9,12,17,18);(3,5)(4,9)(6,13)(7,15)(8,14)(10,18)(11,17)(12,16)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 16459);