Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C73 · Schur multiplier (H2) = C210 |
| direct product | C2×C73×#2888 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C22×C22⋊C4 |
| commutator | C22 | C24×C4×C73 |
| Frattini, cosocle | C2×C4 | C25×C73 |
| prime radical | C22×H(5,𝔽2)×C73 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×#41×C73 ◁ G | |
5114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 5 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);(3,4,6,10,7,8,13,15)(5,9,12,16,11,14,17,18);(4,8)(6,13)(9,14)(12,17);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18);(5,11)(9,14)(12,17)(16,18)5986 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53062);