Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22 |
| direct product | C73×#13270 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C73 | #124 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C2×C73 | #1985 |
| prime radical | C23×C73 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C73 ◁ C22×C8×C73 ◁ G | |
316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,6,12,7,11,23,26,8,13,24,30,5,19,15,31)(3,14,18,22,17,27,32,20,16,9,25,21,10,28,29,4)(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);(1,3,15,29,5,10,24,25,8,16,23,32,7,17,6,18)(2,9,26,4,19,27,12,21,13,14,31,20,11,28,30,22);(1,4,5,21,8,20,7,22)(2,10,11,3,13,17,19,16)(6,14,15,28,24,9,23,27)(12,29,26,18,30,32,31,25)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6258);