Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#12941 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #447 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| prime radical | C23×C73 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ #405×C73 ◁ G | |
388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,4,3)(6,12)(7,17)(8,9)(10,22)(13,29)(14,15)(16,28)(18,19)(20,36)(21,25)(23,24)(26,35)(27,31)(32,33)(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);(1,3,4,2)(5,6)(7,13)(8,21)(9,12)(10,25)(11,16)(14,23)(15,22)(17,30)(18,27)(19,29)(20,33)(24,28)(26,32)(31,35)(34,36);(5,7,11,20)(6,13,16,33)(8,17,23,36)(9,18,24,35)(10,26,22,19)(12,27,28,31)(14,34,21,30)(15,29,25,32)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5929);