Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C73 · Schur multiplier (H2) = C210 |
| direct product | C73×#60086 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C22×D4 |
| commutator | C4 | C24×C4×C73 |
| Frattini, cosocle | C2×C4 | C25×C73 |
| socle | C22×C73 | C23×D4 |
| prime radical | C2×#585×C73 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C22×#41×C73 ◁ G | |
4874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 5 generators on 109 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,14)(10,23)(11,15)(12,18)(17,30)(19,26)(20,28)(21,29)(22,33)(24,27)(25,32)(31,36)(34,35)(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);(5,7)(6,12)(8,17)(9,18)(10,24)(11,19)(13,25)(14,26)(15,27)(16,23)(20,31)(21,32)(22,34)(28,35)(29,36)(30,33);(5,8)(6,13)(7,17)(9,21)(10,22)(11,20)(12,25)(14,29)(15,28)(16,30)(18,32)(19,31)(23,33)(24,34)(26,36)(27,35);(8,20)(13,28)(17,31)(21,22)(25,35)(29,33)(30,36)(32,34);(5,9,11,10)(6,14,15,23)(7,18,19,24)(8,21,20,22)(12,26,27,16)(13,29,28,33)(17,32,31,34)(25,36,35,30)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53074);