Finite Group DatabasebetaFinite Group Databasebeta| order | 17739 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C73 · Schur multiplier (H2) = C32 |
| direct product | C73×#6895 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C73 | C33 |
| commutator, Frattini, cosocle | C32 | C33×C73 |
| prime radical | C33×C73 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C73 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 5913 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,4,3,6,8,7,5,9)(10,11,14)(12,16,22)(13,17,21)(15,23,20)(18,25,30)(19,26,29)(24,31,28)(27,33,35)(32,36,34)(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);(11,15,17)(14,21,23)(16,24,26)(22,29,31)(25,32,33)(30,35,36);(2,5,6)(4,8,9)(10,12,18,13,19,27,20,28,34)(11,16,25,17,26,33,15,24,32)(14,22,30,21,29,35,23,31,36)3723 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17739, 42);