Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C73 · Schur multiplier (H2) = C25 |
| direct product | C2×C8×Q16×C73 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C73 | D4 |
| commutator | C4 | C23×C8×C73 |
| Frattini, cosocle | C42 | C24×C73 |
| socle, prime radical | C23×C73 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C73 ◁ C2×C4×C8×C73 ◁ G | |
734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 18688 |
A minimal faithful permutation representation — 4 generators on 99 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(3,4,5,7,6,8,9,10);(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19)8176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 10832);