Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#13285 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C2×D8 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | C2×#102 |
| prime radical | C23×C73 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,20,16,21,14,31)(11,22,26,38,27,39,25,34)(12,19,13,30,29,33,28,18)(17,32,35,40,36,23,37,24)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,5)(6,7)(9,11,13,24,15,26,29,32,16,27,28,40,14,25,12,23)(10,17,33,39,31,37,30,38,21,36,19,22,20,35,18,34);(1,3,4,8)(2,6,5,7)(9,12,14,28,16,29,15,13)(10,18,20,19,21,30,31,33)(11,23,25,40,27,32,26,24)(17,34,35,22,36,38,37,39)5548 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6273);