Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C24 |
| direct product | C73×#12982 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C73 | #120 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C2×C73 | #1869 |
| prime radical | #532×C73 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C73 ◁ C22×C8×C73 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,6,12,7,13,25,31,8,11,26,30,5,10,15,28)(3,14,18,23,16,29,32,4,17,20,24,19,22,9,27,21)(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);(1,3,15,32,5,22,26,18,8,17,25,27,7,16,6,24)(2,9,28,23,10,20,30,21,11,29,31,19,13,14,12,4);(1,4)(2,3)(5,23)(6,14)(7,19)(8,21)(9,15)(10,22)(11,17)(12,24)(13,16)(18,30)(20,26)(25,29)(27,31)(28,32)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5970);