Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C210 |
| direct product | C73×#60401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C73 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C22×D4×C73 ◁ G | |
4854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 5 generators on 105 points:
(1,2,6,10)(3,7,13,19)(4,8,15,21)(5,9,16,22)(11,23,25,17)(12,18,26,29)(14,20,27,30)(24,31,32,28)(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)(2,7)(4,11)(5,12)(6,13)(8,17)(9,18)(10,19)(14,24)(15,25)(16,26)(20,28)(21,23)(22,29)(27,32)(30,31);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,20)(10,21)(12,24)(13,25)(16,27)(18,28)(19,23)(22,30)(26,32)(29,31);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,20)(10,22)(11,24)(13,26)(15,27)(17,28)(19,29)(21,30)(23,31)(25,32);(5,16)(9,22)(12,26)(14,27)(18,29)(20,30)(24,32)(28,31)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53389);