Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C24 |
| direct product | C73×#13022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #447 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| prime radical | C2×#509×C73 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ #405×C73 ◁ G | |
494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 109 points:
(2,4)(5,6,8,14,25,29,9,15,11,17,26,30,10,16,23,12)(7,18,19,13,27,31,24,35,22,34,36,33,21,28,20,32)(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)(6,12)(8,23)(9,26)(10,25)(13,28)(14,16)(15,17)(18,32)(19,20)(21,27)(24,36)(29,30)(31,33)(34,35);(5,7)(6,13)(8,24)(9,19)(10,27)(11,22)(12,28)(14,34)(15,31)(16,35)(17,33)(18,30)(20,26)(21,25)(23,36)(29,32)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6010);