Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C73 · Schur multiplier (H2) = C22 |
| direct product | C2×C73×#1422 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C73 | D4 |
| commutator | C4 | C22×C16×C73 |
| Frattini, cosocle | C2×C16 | C23×C73 |
| socle | C22×C73 | C4⋊C16 |
| prime radical | C23×C73 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C73 ◁ C22×C16×C73 ◁ G | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 107 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(3,4,7,13,8,14,22,29,9,12,23,27,19,28,32,34,10,15,24,30,17,11,25,26,6,20,21,33,5,16,18,31);(3,5,6,17,10,19,9,8)(4,11,12,16,15,14,20,28)(7,18,21,25,24,32,23,22)(13,26,27,31,30,29,33,34)8176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6617);