Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#12850 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #117 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | #432 |
| prime radical | C24×C73 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×#97×C73 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2)(3,4)(5,6,9,16,10,13,25,29,11,14,24,30,8,15,22,31)(7,17,20,32,21,26,34,35,18,12,23,27,19,28,33,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(5,7)(6,12)(8,19)(9,23)(10,21)(11,18)(13,28)(14,17)(15,26)(16,32)(20,24)(22,34)(25,33)(27,30)(29,35)(31,36);(2,4)(6,13,14,15)(7,18)(9,24)(12,26,17,28)(16,31,30,29)(19,21)(22,25)(27,36,32,35)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5838);