Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#8058 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C73 |
| Frattini, cosocle | C23×C4 | C23×C73 |
| socle | C22×C73 | C2×#95 |
| prime radical | C24×C73 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C22×C42×C73 ◁ G | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(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);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,19)(18,22,24,25)(20,28)(21,27)(23,31,32,30)(26,29);(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1046);