Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C29 |
| direct product | C73×#60424 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| prime radical | C73×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×#117×C73 ◁ G | |
2674 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)(7,11)(8,18)(9,21)(10,13)(12,19)(14,23)(15,29)(16,25)(17,33)(20,28)(22,27)(24,31)(26,32)(30,36)(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);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,25)(14,26)(18,29)(19,31)(20,30)(21,33)(22,34)(23,32)(27,35)(28,36);(5,8,10,20)(6,12,13,27)(7,15,16,30)(9,22,23,19)(11,24,25,35)(14,28,21,18)(17,34,32,31)(26,36,33,29);(7,16)(11,25)(15,30)(17,32)(24,35)(26,33)(29,36)(31,34);(6,13)(11,25)(12,27)(14,21)(18,28)(24,35)(26,33)(29,36)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53412);