Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C27 |
| direct product | C73×#20944 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C22×#41×C73 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
1702 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)(5,6,10,14,11,16,9,15)(7,17,19,12,21,27,20,28)(8,22,24,13,25,29,23,30)(18,26,32,35,34,36,33,31)(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)(8,18)(9,19)(10,20)(11,21)(13,26)(14,17)(15,27)(16,28)(22,35)(23,32)(24,33)(25,34)(29,31)(30,36);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,32)(20,33)(21,34)(27,35)(28,36);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,17,28,27)(13,29,30,22)(18,32,34,33)(26,31,36,35)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 13932);