Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C27 |
| direct product | C73×#20962 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator | C23 | C23×C4×C73 |
| Frattini, cosocle | C24 | C24×C73 |
| prime radical | C22×#41×C73 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(2,4)(5,6)(7,11)(8,18,20,12)(9,21)(10,14)(13,22)(15,23,28,31)(16,29)(17,25)(19,26,33,32)(24,30)(27,35,36,34)(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);(1,2,3,4)(5,7,10,17)(6,11,14,25)(8,15,20,28)(9,16,22,30)(12,23,18,31)(13,24,21,29)(19,27,33,36)(26,34,32,35);(5,8)(6,12)(7,15)(9,19)(10,20)(11,23)(13,26)(14,18)(16,27)(17,28)(21,32)(22,33)(24,34)(25,31)(29,35)(30,36);(5,9,10,22)(6,13,14,21)(7,16,17,30)(8,19,20,33)(11,24,25,29)(12,26,18,32)(15,27,28,36)(23,34,31,35)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 13950);