Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#13262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #459 |
| 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 ◁ C2×C4×C8×C73 ◁ G | |
462 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)(5,6,9,14,10,25,23,32,12,16,22,31,11,15,8,21)(7,17,18,28,26,33,36,30,20,34,35,29,19,27,24,13)(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)(5,7,12,20)(6,13,16,30)(8,18,23,35)(9,24,22,36)(10,19,11,26)(14,27,31,33)(15,28,25,29)(17,32,34,21);(5,8,11,22,12,23,10,9)(6,14,25,32,16,31,15,21)(7,18,26,36,20,35,19,24)(13,27,29,34,30,33,28,17)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6250);