Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#22896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | D42 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C2×H(5,𝔽2)×C73 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ G | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(1,2,4,3)(5,6)(7,18)(8,14)(9,16)(10,15)(11,17)(12,21)(13,23)(19,27)(20,34)(22,33)(24,32)(25,29)(26,30)(28,36)(31,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(1,3,4,2)(5,7)(6,12)(8,19)(9,25)(10,21)(11,22)(13,27)(14,31)(15,33)(16,18)(17,29)(20,23)(24,36)(26,35)(28,30)(32,34);(5,8,11,24)(6,13,17,30)(7,19,22,36)(9,23,10,26)(12,27,29,28)(14,16,32,15)(18,34,33,31)(20,21,35,25);(5,8)(6,14)(7,20)(9,26)(10,23)(11,24)(12,28)(13,15)(16,30)(17,32)(18,34)(19,25)(21,36)(22,35)(27,29)(31,33)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 15884);