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×#13259 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | #459 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C2×C73 | #1982 |
| prime radical | C73×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C22×C4×C73 ◁ C2×C4×C8×C73 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,6,13,7,12,22,31,8,14,21,10,5,23,4,11)(3,15,19,29,18,30,17,26,20,25,16,9,24,27,32,28)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3,7,24,8,20,5,18)(2,9,12,26,14,29,23,28)(4,16,6,17,22,19,21,32)(10,25,11,30,13,15,31,27);(1,4,5,21,8,22,7,6)(2,10,12,11,14,13,23,31)(3,16,18,32,20,19,24,17)(9,25,26,30,29,15,28,27)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6247);