Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C25 |
| direct product | C2×C73×#1841 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #117 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C22×C73 | #383 |
| prime radical | C24×C73 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C22×M16×C73 ◁ G | |
730 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)(5,13,16,17)(6,9)(7,11)(8,12)(10,15,18,14);(4,9,12,11)(10,17,18,13);(3,5,6,14,8,16,7,15)(4,10,11,13,12,18,9,17)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 13763);