Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#12012 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C73 | D4 |
| commutator | C4 | C22×C16×C73 |
| Frattini, cosocle | C4×C8 | C23×C73 |
| socle | C22×C73 | C2×C4⋊C8 |
| prime radical | C23×C73 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C73 ◁ C2×C4×C8×C73 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)8176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5000);