Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22 |
| direct product | C73×#13187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #476 |
| 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 | |
294 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)(7,17,21,35)(8,23)(9,13)(10,28)(11,16)(12,19,30,18)(14,25)(15,24)(20,33,22,34)(26,31,36,29)(27,32)(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);(5,7,10,20,11,21,24,22)(6,12,15,29,16,30,28,31)(8,18,9,26,25,19,27,36)(13,17,23,34,32,35,14,33);(1,2,3,4)(5,8,24,27,11,25,10,9)(6,13,28,14,16,32,15,23)(7,18,22,36,21,19,20,26)(12,17,31,33,30,35,29,34)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6175);