Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#13265 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C73 | C2×D4 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | C2×C22⋊C8 |
| prime radical | C23×C73 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C73 ◁ C2×C4×C8×C73 ◁ G | |
338 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)(9,10,13,18,14,19,28,36,15,20,29,37,12,17,22,31)(11,21,24,38,23,33,39,35,25,34,27,40,26,16,30,32)(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)(9,11,22,39,12,26,29,24,15,25,28,30,14,23,13,27)(10,16,31,38,17,34,37,32,20,33,36,40,19,21,18,35);(1,3,4,8)(2,6,5,7)5548 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6253);