Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#12618 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C2×C73 | #1752 |
| prime radical | C2×#41×C73 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C2×M32×C73 ◁ G | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,4,9,5,10,15,23,6,7,14,18,16,19,28,30)(3,8,12,21,11,22,24,32,13,17,27,29,25,20,26,31)(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);(2,7)(3,11,13,25)(4,14)(8,20,17,22)(10,19)(12,26,27,24)(15,28)(21,32,29,31);(1,3,4,12,5,11,15,24,6,13,14,27,16,25,28,26)(2,8,9,21,10,22,23,32,7,17,18,29,19,20,30,31)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5606);