Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C27 |
| direct product | D4×#99×C73 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C73 |
| Frattini, cosocle | C24 | C24×C73 |
| prime radical | C2×D42×C73 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×D4×C73 ◁ G | |
2102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(2,4);(5,6)(7,11,12,9)(8,10);(5,7)(6,9)(8,12)(10,11)4015 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 15392);