Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22 |
| direct product | C73×#8606 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #121 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| socle | C22×C73 | #380 |
| prime radical | C2×#41×C73 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×C42×C73 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,24,21,25,30,23)(19,26,28,22,29,31,32,27)(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)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,20,28,21,29,30,32)(18,22,23,26,25,27,24,31);(3,9)(6,12)(10,15)(13,16)(17,18)(19,27)(20,23)(21,25)(22,29)(24,30)(26,32)(28,31)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1594);