Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#8125 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#95 |
| commutator | C23 | C2×C42×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| prime radical | C24×C73 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C42×C73 ◁ G | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,4,9,6,8,5,10)(3,11,12,16,13,7,14,15)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30)(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);(1,3,5,12,6,13,4,14)(2,7,10,15,8,11,9,16)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29);(2,8)(7,11)(9,10)(15,16)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1113);