Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C29 |
| direct product | C73×#60427 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| prime radical | C2×#575×C73 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×#117×C73 ◁ G | |
2678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 5 generators on 109 points:
(2,4)(5,6,11,16)(7,12,18,26)(8,21,23,30)(9,15,10,14)(13,24,29,22)(17,31,32,36)(19,28,20,27)(25,34,35,33)(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)(6,12)(8,17)(9,19)(10,20)(11,18)(13,25)(14,27)(15,28)(16,26)(21,31)(22,33)(23,32)(24,34)(29,35)(30,36);(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,21)(15,30)(16,29)(18,32)(19,33)(20,34)(26,35)(27,31)(28,36);(7,18)(12,26)(17,32)(19,20)(25,35)(27,28)(31,36)(33,34);(5,9,11,10)(6,14,16,15)(7,19,18,20)(8,22,23,24)(12,27,26,28)(13,21,29,30)(17,33,32,34)(25,31,35,36)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53415);