Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#12624 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | C2×#99 |
| commutator | C23 | C2×C42×C73 |
| Frattini, cosocle | C2×#41 | C23×C73 |
| socle | C2×C73 | C4×#99 |
| prime radical | #525×C73 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C2×C8×C73 ◁ C2×#131×C73 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,6,4,7,10,21,17)(3,11,14,26,15,24,28,29)(5,9,19,16,20,25,32,30)(8,22,12,18,23,31,27,13)(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)(2,8)(4,12)(5,18,20,13)(6,14)(7,15)(9,24,25,11)(10,23)(16,29,30,26)(17,27)(19,31,32,22)(21,28);(1,4,6,10,7,17,21,2)(3,12,14,23,15,27,28,8)(5,16,19,25,20,30,32,9)(11,13,26,22,24,18,29,31)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5612);