Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C22×C42 |
| direct product | C73×#22619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C73 | C22×D4 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C24×C73 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C4×C22⋊C4×C73 ◁ G | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 101 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,17,19,22,16,21)(15,23,24,20,26,27,25,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,27)(22,28);(6,10)(9,12)(13,16,19,18)(14,21,22,17)(15,24,26,25)(20,27,28,23);(13,17,18,22,19,21,16,14)(15,20,24,27,26,28,25,23)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 15607);