Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C23×C4 |
| direct product | C73×#19730 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C73 |
| prime radical | C73×#3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 105 points:
(1,2,6,10)(3,11,13,8)(4,7,14,16)(5,15,12,9)(19,25)(20,26)(22,28)(24,29)(27,31)(30,32)(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);(1,3,12,14,6,13,5,4)(2,7,15,8,10,16,9,11)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,4,5,13,6,14,12,3)(2,8,9,7,10,11,15,16)(17,19,26,31,21,27,20,25)(18,22,29,32,23,30,24,28);(3,13)(4,14)(7,16)(8,11)(18,23)(22,30)(24,29)(28,32)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 12718);